assignment 9 matrices

7.5 Matrices and Matrix Operations

Learning objectives.

In this section, you will:

• Find the sum and difference of two matrices.
• Find scalar multiples of a matrix.
• Find the product of two matrices.

Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season. Table 1 shows the needs of both teams.

A goal costs $300; a ball costs $10; and a jersey costs $30. How can we find the total cost for the equipment needed for each team? In this section, we discover a method in which the data in the soccer equipment table can be displayed and used for calculating other information. Then, we will be able to calculate the cost of the equipment.

Finding the Sum and Difference of Two Matrices

To solve a problem like the one described for the soccer teams, we can use a matrix , which is a rectangular array of numbers. A row in a matrix is a set of numbers that are aligned horizontally. A column in a matrix is a set of numbers that are aligned vertically. Each number is an entry , sometimes called an element, of the matrix. Matrices (plural) are enclosed in [ ] or ( ), and are usually named with capital letters. For example, three matrices named A , B , A , B , and C C are shown below.

Describing Matrices

A matrix is often referred to by its size or dimensions: m × n m × n indicating m m rows and n n columns. Matrix entries are defined first by row and then by column. For example, to locate the entry in matrix A A identified as a i j , a i j , we look for the entry in row i , i , column j . j . In matrix A ,   A ,   shown below, the entry in row 2, column 3 is a 23 . a 23 .

A square matrix is a matrix with dimensions n × n , n × n , meaning that it has the same number of rows as columns. The 3 × 3 3 × 3 matrix above is an example of a square matrix.

A row matrix is a matrix consisting of one row with dimensions 1 × n . 1 × n .

A column matrix is a matrix consisting of one column with dimensions m × 1. m × 1.

A matrix may be used to represent a system of equations. In these cases, the numbers represent the coefficients of the variables in the system. Matrices often make solving systems of equations easier because they are not encumbered with variables. We will investigate this idea further in the next section, but first we will look at basic matrix operations .

A matrix is a rectangular array of numbers that is usually named by a capital letter: A , B , C , A , B , C , and so on. Each entry in a matrix is referred to as a i j , a i j , such that i i represents the row and j j represents the column. Matrices are often referred to by their dimensions: m × n m × n indicating m m rows and n n columns.

Finding the Dimensions of the Given Matrix and Locating Entries

Given matrix A : A :

• ⓐ What are the dimensions of matrix A ? A ?
• ⓑ What are the entries at a 31 a 31 and a 22 ? a 22 ? A = [ 2 1 0 2 4 7 3 1 − 2 ] A = [ 2 1 0 2 4 7 3 1 − 2 ]
• ⓐ The dimensions are 3 × 3 3 × 3 because there are three rows and three columns.
• ⓑ Entry a 31 a 31 is the number at row 3, column 1, which is 3. The entry a 22 a 22 is the number at row 2, column 2, which is 4. Remember, the row comes first, then the column.

Adding and Subtracting Matrices

We use matrices to list data or to represent systems. Because the entries are numbers, we can perform operations on matrices. We add or subtract matrices by adding or subtracting corresponding entries.

In order to do this, the entries must correspond. Therefore, addition and subtraction of matrices is only possible when the matrices have the same dimensions . We can add or subtract a 3 × 3 3 × 3 matrix and another 3 × 3 3 × 3 matrix, but we cannot add or subtract a 2 × 3 2 × 3 matrix and a 3 × 3 3 × 3 matrix because some entries in one matrix will not have a corresponding entry in the other matrix.

Given matrices A A and B B of like dimensions, addition and subtraction of A A and B B will produce matrix C C or matrix D D of the same dimension.

Matrix addition is commutative.

It is also associative.

Finding the Sum of Matrices

Find the sum of A A and B , B , given

Add corresponding entries.

Adding Matrix A and Matrix B

Find the sum of A A and B . B .

Add corresponding entries. Add the entry in row 1, column 1, a 11 , a 11 , of matrix A A to the entry in row 1, column 1, b 11 , b 11 , of B . B . Continue the pattern until all entries have been added.

Finding the Difference of Two Matrices

Find the difference of A A and B . B .

We subtract the corresponding entries of each matrix.

Finding the Sum and Difference of Two 3 x 3 Matrices

Given A A and B : B :

• ⓐ Find the sum.
• ⓑ Find the difference.
• ⓐ Add the corresponding entries. A + B = [ 2 − 10 − 2 14 12 10 4 − 2 2 ] + [ 6 10 − 2 0 − 12 − 4 − 5 2 − 2 ] = [ 2 + 6 − 10 + 10 − 2 − 2 14 + 0 12 − 12 10 − 4 4 − 5 − 2 + 2 2 − 2 ] = [ 8 0 − 4 14 0 6 − 1 0 0 ] A + B = [ 2 − 10 − 2 14 12 10 4 − 2 2 ] + [ 6 10 − 2 0 − 12 − 4 − 5 2 − 2 ] = [ 2 + 6 − 10 + 10 − 2 − 2 14 + 0 12 − 12 10 − 4 4 − 5 − 2 + 2 2 − 2 ] = [ 8 0 − 4 14 0 6 − 1 0 0 ]
• ⓑ Subtract the corresponding entries. A − B = [ 2 −10 −2 14 12 10 4 −2 2 ] − [ 6 10 −2 0 −12 −4 −5 2 −2 ] = [ 2 − 6 −10 − 10 −2 + 2 14 − 0 12 + 12 10 + 4 4 + 5 −2 − 2 2 + 2 ] = [ −4 −20 0 14 24 14 9 −4 4 ] A − B = [ 2 −10 −2 14 12 10 4 −2 2 ] − [ 6 10 −2 0 −12 −4 −5 2 −2 ] = [ 2 − 6 −10 − 10 −2 + 2 14 − 0 12 + 12 10 + 4 4 + 5 −2 − 2 2 + 2 ] = [ −4 −20 0 14 24 14 9 −4 4 ]

Add matrix A A and matrix B . B .

Finding Scalar Multiples of a Matrix

Besides adding and subtracting whole matrices, there are many situations in which we need to multiply a matrix by a constant called a scalar. Recall that a scalar is a real number quantity that has magnitude, but not direction. For example, time, temperature, and distance are scalar quantities. The process of scalar multiplication involves multiplying each entry in a matrix by a scalar. A scalar multiple is any entry of a matrix that results from scalar multiplication.

Consider a real-world scenario in which a university needs to add to its inventory of computers, computer tables, and chairs in two of the campus labs due to increased enrollment. They estimate that 15% more equipment is needed in both labs. The school’s current inventory is displayed in Table 2 .

Converting the data to a matrix, we have

To calculate how much computer equipment will be needed, we multiply all entries in matrix C C by 0.15.

We must round up to the next integer, so the amount of new equipment needed is

Adding the two matrices as shown below, we see the new inventory amounts.

Thus, Lab A will have 18 computers, 19 computer tables, and 19 chairs; Lab B will have 32 computers, 40 computer tables, and 40 chairs.

Scalar Multiplication

Scalar multiplication involves finding the product of a constant by each entry in the matrix. Given

the scalar multiple c A c A is

Scalar multiplication is distributive. For the matrices A , B , A , B , and C C with scalars a a and b , b ,

Multiplying the Matrix by a Scalar

Multiply matrix A A by the scalar 3.

Multiply each entry in A A by the scalar 3.

Given matrix B , B , find −2 B −2 B where

Finding the Sum of Scalar Multiples

Find the sum 3 A + 2 B . 3 A + 2 B .

First, find 3 A , 3 A , then 2 B . 2 B .

Now, add 3 A + 2 B . 3 A + 2 B .

Finding the Product of Two Matrices

In addition to multiplying a matrix by a scalar, we can multiply two matrices. Finding the product of two matrices is only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix. If A A is an m × r m × r matrix and B B is an r × n r × n matrix, then the product matrix A B A B is an m × n m × n matrix. For example, the product A B A B is possible because the number of columns in A A is the same as the number of rows in B . B . If the inner dimensions do not match, the product is not defined.

We multiply entries of A A with entries of B B according to a specific pattern as outlined below. The process of matrix multiplication becomes clearer when working a problem with real numbers.

To obtain the entries in row i i of A B , A B , we multiply the entries in row i i of A A by column j j in B B and add. For example, given matrices A A and B , B , where the dimensions of A A are 2 × 3 2 × 3 and the dimensions of B B are 3 × 3 , 3 × 3 , the product of A B A B will be a 2 × 3 2 × 3 matrix.

Multiply and add as follows to obtain the first entry of the product matrix A B . A B .

• To obtain the entry in row 1, column 1 of A B , A B , multiply the first row in A A by the first column in B , B , and add. [ a 11 a 12 a 13 ] [ b 11 b 21 b 31 ] = a 11 ⋅ b 11 + a 12 ⋅ b 21 + a 13 ⋅ b 31 [ a 11 a 12 a 13 ] [ b 11 b 21 b 31 ] = a 11 ⋅ b 11 + a 12 ⋅ b 21 + a 13 ⋅ b 31
• To obtain the entry in row 1, column 2 of A B , A B , multiply the first row of A A by the second column in B , B , and add. [ a 11 a 12 a 13 ] [ b 12 b 22 b 32 ] = a 11 ⋅ b 12 + a 12 ⋅ b 22 + a 13 ⋅ b 32 [ a 11 a 12 a 13 ] [ b 12 b 22 b 32 ] = a 11 ⋅ b 12 + a 12 ⋅ b 22 + a 13 ⋅ b 32
• To obtain the entry in row 1, column 3 of A B , A B , multiply the first row of A A by the third column in B , B , and add. [ a 11 a 12 a 13 ] [ b 13 b 23 b 33 ] = a 11 ⋅ b 13 + a 12 ⋅ b 23 + a 13 ⋅ b 33 [ a 11 a 12 a 13 ] [ b 13 b 23 b 33 ] = a 11 ⋅ b 13 + a 12 ⋅ b 23 + a 13 ⋅ b 33

We proceed the same way to obtain the second row of A B . A B . In other words, row 2 of A A times column 1 of B ; B ; row 2 of A A times column 2 of B ; B ; row 2 of A A times column 3 of B . B . When complete, the product matrix will be

Properties of Matrix Multiplication

For the matrices A , B , A , B , and C C the following properties hold.

• Matrix multiplication is associative: ( A B ) C = A ( B C ) . ( A B ) C = A ( B C ) .
• Matrix multiplication is distributive: C ( A + B ) = C A + C B , ( A + B ) C = A C + B C . C ( A + B ) = C A + C B , ( A + B ) C = A C + B C .

Note that matrix multiplication is not commutative.

Multiplying Two Matrices

Multiply matrix A A and matrix B . B .

First, we check the dimensions of the matrices. Matrix A A has dimensions 2 × 2 2 × 2 and matrix B B has dimensions 2 × 2. 2 × 2. The inner dimensions are the same so we can perform the multiplication. The product will have the dimensions 2 × 2. 2 × 2.

We perform the operations outlined previously.

• ⓐ Find A B . A B .
• ⓑ Find B A . B A .
• ⓐ As the dimensions of A A are 2 × 3 2 × 3 and the dimensions of B B are 3 × 2 , 3 × 2 , these matrices can be multiplied together because the number of columns in A A matches the number of rows in B . B . The resulting product will be a 2 × 2 2 × 2 matrix, the number of rows in A A by the number of columns in B . B . A B = [ −1 2 3 4 0 5 ]    [ 5 −1 − 4 0 2 3 ] = [ −1 ( 5 ) + 2 ( −4 ) + 3 ( 2 ) −1 ( −1 ) + 2 ( 0 ) + 3 ( 3 ) 4 ( 5 ) + 0 ( −4 ) + 5 ( 2 ) 4 ( −1 ) + 0 ( 0 ) + 5 ( 3 ) ] = [ −7 10 30 11 ] A B = [ −1 2 3 4 0 5 ]    [ 5 −1 − 4 0 2 3 ] = [ −1 ( 5 ) + 2 ( −4 ) + 3 ( 2 ) −1 ( −1 ) + 2 ( 0 ) + 3 ( 3 ) 4 ( 5 ) + 0 ( −4 ) + 5 ( 2 ) 4 ( −1 ) + 0 ( 0 ) + 5 ( 3 ) ] = [ −7 10 30 11 ]
• ⓑ The dimensions of B B are 3 × 2 3 × 2 and the dimensions of A A are 2 × 3. 2 × 3. The inner dimensions match so the product is defined and will be a 3 × 3 3 × 3 matrix. B A = [ 5 −1 −4 0 2 3 ]    [ −1 2 3 4 0 5 ] = [ 5 ( −1 ) + −1 ( 4 ) 5 ( 2 ) + −1 ( 0 ) 5 ( 3 ) + −1 ( 5 ) −4 ( −1 ) + 0 ( 4 ) −4 ( 2 ) + 0 ( 0 ) −4 ( 3 ) + 0 ( 5 ) 2 ( −1 ) + 3 ( 4 ) 2 ( 2 ) + 3 ( 0 ) 2 ( 3 ) + 3 ( 5 ) ] = [ −9 10 10 4 −8 −12 10 4 21 ] B A = [ 5 −1 −4 0 2 3 ]    [ −1 2 3 4 0 5 ] = [ 5 ( −1 ) + −1 ( 4 ) 5 ( 2 ) + −1 ( 0 ) 5 ( 3 ) + −1 ( 5 ) −4 ( −1 ) + 0 ( 4 ) −4 ( 2 ) + 0 ( 0 ) −4 ( 3 ) + 0 ( 5 ) 2 ( −1 ) + 3 ( 4 ) 2 ( 2 ) + 3 ( 0 ) 2 ( 3 ) + 3 ( 5 ) ] = [ −9 10 10 4 −8 −12 10 4 21 ]

Notice that the products A B A B and B A B A are not equal.

This illustrates the fact that matrix multiplication is not commutative.

Is it possible for AB to be defined but not BA ?

Yes, consider a matrix A with dimension 3 × 4 3 × 4 and matrix B with dimension 4 × 2. 4 × 2. For the product AB the inner dimensions are 4 and the product is defined, but for the product BA the inner dimensions are 2 and 3 so the product is undefined.

Using Matrices in Real-World Problems

Let’s return to the problem presented at the opening of this section. We have Table 3 , representing the equipment needs of two soccer teams.

We are also given the prices of the equipment, as shown in Table 4 .

We will convert the data to matrices. Thus, the equipment need matrix is written as

The cost matrix is written as

We perform matrix multiplication to obtain costs for the equipment.

The total cost for equipment for the Wildcats is $2,520, and the total cost for equipment for the Mud Cats is $3,840.

Given a matrix operation, evaluate using a calculator.

• Save each matrix as a matrix variable [ A ] , [ B ] , [ C ] , ... [ A ] , [ B ] , [ C ] , ...
• Enter the operation into the calculator, calling up each matrix variable as needed.
• If the operation is defined, the calculator will present the solution matrix; if the operation is undefined, it will display an error message.

Using a Calculator to Perform Matrix Operations

Find A B − C A B − C given

On the matrix page of the calculator, we enter matrix A A above as the matrix variable [ A ] , [ A ] , matrix B B above as the matrix variable [ B ] , [ B ] , and matrix C C above as the matrix variable [ C ] . [ C ] .

On the home screen of the calculator, we type in the problem and call up each matrix variable as needed.

The calculator gives us the following matrix.

Access these online resources for additional instruction and practice with matrices and matrix operations.

• Dimensions of a Matrix
• Matrix Addition and Subtraction
• Matrix Operations
• Matrix Multiplication

7.5 Section Exercises

Can we add any two matrices together? If so, explain why; if not, explain why not and give an example of two matrices that cannot be added together.

Can we multiply any column matrix by any row matrix? Explain why or why not.

Can both the products A B A B and B A B A be defined? If so, explain how; if not, explain why.

Can any two matrices of the same size be multiplied? If so, explain why, and if not, explain why not and give an example of two matrices of the same size that cannot be multiplied together.

Does matrix multiplication commute? That is, does A B = B A ? A B = B A ? If so, prove why it does. If not, explain why it does not.

For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined.

A + B A + B

C + D C + D

A + C A + C

B − E B − E

C + F C + F

D − B D − B

For the following exercises, use the matrices below to perform scalar multiplication.

1 2 C 1 2 C

100 D 100 D

For the following exercises, use the matrices below to perform matrix multiplication.

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed.

A + B − C A + B − C

4 A + 5 D 4 A + 5 D

2 C + B 2 C + B

3 D + 4 E 3 D + 4 E

C −0.5 D C −0.5 D

100 D −10 E 100 D −10 E

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: A 2 = A ⋅ A A 2 = A ⋅ A )

B 2 A 2 B 2 A 2

A 2 B 2 A 2 B 2

( A B ) 2 ( A B ) 2

( B A ) 2 ( B A ) 2

( A B ) C ( A B ) C

A ( B C ) A ( B C )

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution.

A B C A B C

For the following exercises, use the matrix below to perform the indicated operation on the given matrix.

Using the above questions, find a formula for B n . B n . Test the formula for B 201 B 201 and B 202 , B 202 , using a calculator.

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Free Mathematics Tutorials

Matrices with Examples and Questions with Solutions

Examples and questions on matrices along with their solutions are presented .

Page Content

Definition of a matrix, matrix entry (or element), square matrix, identity matrix, diagonal matrix, triangular matrix, transpose of a matrix, symmetric matrix, questions on matrices: part a, questions on matrices: part b, solutions to the questions in part a, solutions to the questions in part b.

The following are examples of matrices (plural of matrix ).

Example 1 The following matrix has 3 rows and 6 columns.

The entry (or element) in a row i and column j of a matrix A (capital letter A) is denoted by the symbol \((A)_{ij} \) or \( a_{ij} \) (small letter a).

In the matrix A shown below, \(a_{11} = 5 \), \(a_{12} = 2 \), etc ... or \( (A)_{11} = 5 \), \( (A)_{12} = 2 \), etc ... \[ \textbf{A} = \begin{bmatrix} 5 & 2 & 7 & -3 \\ -9 & -2 & -7 & 11\\ \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ \end{bmatrix} \]

A square matrix has the number of rows equal to the number of columns.

For each matrix below, determine the order and state whether it is a square matrix. \[ a) \begin{bmatrix} -1 & 1 & 0 & 3 \\ 4 & -3 & -7 & -9\\ \end{bmatrix} \;\;\;\; b) \begin{bmatrix} -6 & 2 & 0 \\ 3 & -3 & 4 \\ -5 & -11 & 9 \end{bmatrix} \;\;\;\; \\ c) \begin{bmatrix} 1 & -2 & 5 & -2 \end{bmatrix} \;\;\;\; d) \begin{bmatrix} -2 & 0 \\ 0 & -3 \end{bmatrix} \;\;\;\; e) \begin{bmatrix} 3 \end{bmatrix} \] Solutions a) order: 2 × 4. Number of rows and columns are not equal therefore not a square matrix. b) order: 3 × 3. Number of rows and columns are equal therefore this matrix is a square matrix. c) order: 1 × 4. Number of rows and columns are not equal therefore not a square matrix. A matrix with one row is called a row matrix (or a row vector). d) order: 2 × 2. Number of rows and columns are equal therefore this is square matrix. e) order: 1 × 1. Number of rows and columns are equal therefore this matrix is a square matrix.

An identity matrix I n is an n×n square matrix with all its element in the diagonal equal to 1 and all other elements equal to zero. Example 4 The following are all identity matrices. \[I_1= \begin{bmatrix} 1 \\ \end{bmatrix} \quad , \quad I_2= \begin{bmatrix} 1 & 0\\ 0& 1 \end{bmatrix} \quad , \quad I_3= \begin{bmatrix} 1 & 0 & 0\\ 0& 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]

A diagonal matrix is a square matrix with all its elements (entries) equal to zero except the elements in the main diagonal from top left to bottom right. \[A = \begin{bmatrix} 6 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 2 \end{bmatrix} \]

An upper triangular matrix is a square matrix with all its elements below the main diagonal equal to zero. Matrix U shown below is an example of an upper triangular matrix. A lower triangular matrix is a square matrix with all its elements above the main diagonal equal to zero. Matrix L shown below is an example of a lower triangular matrix. \(U = \begin{bmatrix} 6 & 2 & -5 \\ 0 & -2 & 7 \\ 0 & 0 & 2 \end{bmatrix} \qquad L = \begin{bmatrix} 6 & 0 & 0 \\ -2 & -2 & 0 \\ 10 & 9 & 2 \end{bmatrix} \)

The transpose of an m×n matrix \( A \) is denoted \( A^T \) with order n×m and defined by \[ (A^T)_{ij} = (A)_{ji} \] Matrix \( A^T \) is obtained by transposing (exchanging) the rows and columns of matrix \( A \). Example 5 \[ \begin{bmatrix} 6 & 0 \\ -2 & -2\\ 10 & 9 \end{bmatrix} ^T = \begin{bmatrix} 6 & -2 & 10 \\ 0 & -2 &9\\ \end{bmatrix} \] Transpose a matrix an even number of times and you get the original matrix: \( ((A)^T)^T = A \). Transpose matrix an odd number of times and you get the transpose matrix: \( (((A)^T)^T)^T = A^T \). The transpose of any square diagonal matrix is the matrix itself. \[ \begin{bmatrix} 3 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 6 \end{bmatrix} ^T = \begin{bmatrix} 3 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 6 \end{bmatrix} \]

A square matrix is symmetric if its elements are such that \( A_{ij} = A_{ji} \) in other words \( A \) is symmetric if \(A = A^T \). Example 6 Symmetric matrices \[ \begin{bmatrix} 4 & -2 & 1 \\ -2 & 5 & 7 \\ 1 & 7 & 8 \end{bmatrix} \]

Given the matrices: \[ A = \begin{bmatrix} -1 & 23 & 10 \\ 0 & -2 & -11 \\ \end{bmatrix} ,\quad B = \begin{bmatrix} -6 & 2 & 10 \\ 3 & -3 & 4 \\ -5 & -11 & 9 \\ 1 & -1 & 9 \end{bmatrix} ,\quad C = \begin{bmatrix} -3 & 2 & 9 & -5 & 7 \end{bmatrix} \\ D = \begin{bmatrix} -2 & 6 \\ -5 & 2\\ \end{bmatrix} ,\quad E = \begin{bmatrix} 3 \end{bmatrix} ,\quad F = \begin{bmatrix} 3 \\ 5 \\ -11 \\ 7 \end{bmatrix} ,\quad G = \begin{bmatrix} -6 & -4 & 23 \\ -4 & -3 & 4 \\ 23 & 4 & 9 \\ \end{bmatrix} \] a) What is the dimension of each matrix? b) Which matrices are square? c) Which matrices are symmetric? d) Which matrix has the entry at row 3 and column 2 equal to -11? e) Which matrices has the entry at row 1 and column 3 equal to 10? f) Which are column matrices? g) Which are row matrices? h) Find \( A^T , C^T , E^T , G^T \).

1) Given the matrices: \[ A = \begin{bmatrix} 23 & 10 \\ 0 & -11 \\ \end{bmatrix} ,\quad B = \begin{bmatrix} -6 & 0 & 0 \\ -1 & -3 & 0 \\ -5 & 3 & -9 \\ \end{bmatrix} ,\quad C = \begin{bmatrix} -3 & 0\\ 0 & 2 \end{bmatrix} \\ ,\quad D = \begin{bmatrix} -7 & 3 & 2 \\ 0 & 2 & 4 \\ 0 & 0 & 9 \\ \end{bmatrix} ,\quad E = \begin{bmatrix} 12 & 0 & 0 \\ 0 & 23 & 0 \\ 0 & 0 & -19\\ \end{bmatrix} \] a) Which of the above matrices are diagonal? b) Which of the above matrices are lower triangular? c) Which of the above matrices are upper triangular?

a) A: 2 × 3, B: 4 × 3, C: 1 × 5, D: 2 × 2, E: 1 × 1, F: 4 × 1, G: 3 × 3, b) D, E and G c) E and G d) B e) A and B f) E and F g) E and C h) \[ A^T = \begin{bmatrix} -1 & 0 \\ 23 & -2 \\ 10 & -11 \end{bmatrix} ,\quad C^T = \begin{bmatrix} -3 \\ 2\\ 9\\-5\\7 \end{bmatrix} ,\quad E^T = \begin{bmatrix} 3 \end{bmatrix} ,\quad G^T = \begin{bmatrix} -6 & -4 & 23\\ -4 & -3 & 4\\ 23 & 4 & 9 \end{bmatrix} \]

More References and links

• Add, Subtract and Scalar Multiply Matrices
• Multiplication and Power of Matrices
• Linear Algebra
• Row Operations and Elementary Matrices
• Matrix (mathematics)
• Matrices Applied to Electric Circuits
• The Inverse of a Square Matrix

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2.5: Application of Matrices in Cryptography

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Learning Objectives

In this section, we will learn to find the inverse of a matrix, if it exists. Later, we will use matrix inverses to solve linear systems. In this section you will learn to

• encode a message using matrix multiplication.
• decode a coded message using the matrix inverse and matrix multiplication

Encryption dates back approximately 4000 years. Historical accounts indicate that the Chinese, Egyptians, Indian, and Greek encrypted messages in some way for various purposes. One famous encryption scheme is called the Caesar cipher, also called a substitution cipher, used by Julius Caesar, involved shifting letters in the alphabet, such as replacing A by C, B by D, C by E, etc, to encode a message. Substitution ciphers are too simple in design to be considered secure today.

In the middle ages, European nations began to use encryption. A variety of encryption methods were used in the US from the Revolutionary War, through the Civil War, and on into to modern times.

Applications of mathematical theory and methods to encryption became widespread in military usage in the 20 th century. The military would encode messages before sending and the recipient would decode the message, in order to send information about military operations in a manner that kept the information safe if the message was intercepted. In World War II, encryption played an important role, as both Allied and Axis powers sent encrypted messages and devoted significant resources to strengthening their own encryption while also trying to break the opposition’s encryption.

In this section we will examine a method of encryption that uses matrix multiplication and matrix inverses. This method, known as the Hill Algorithm, was created by Lester Hill, a mathematics professor who taught at several US colleges and also was involved with military encryption. The Hill algorithm marks the introduction of modern mathematical theory and methods to the field of cryptography.

These days, the Hill Algorithm is not considered a secure encryption method; it is relatively easy to break with modern technology. However, in 1929 when it was developed, modern computing technology did not exist. This method, which we can handle easily with today’s technology, was too cumbersome to use with hand calculations. Hill devised a mechanical encryption machine to help with the mathematics; his machine relied on gears and levers, but never gained widespread use. Hill’s method was considered sophisticated and powerful in its time and is one of many methods influencing techniques in use today. Other encryption methods at that time also utilized special coding machines. Alan Turing, a computer scientist pioneer in the field of artificial intelligence, invented a machine that was able to decrypt messages encrypted by the German Enigma machine, helping to turn the tide of World War II.

With the advent of the computer age and internet communication, the use of encryption has become widespread in communication and in keeping private data secure; it is no longer limited to military uses. Modern encryption methods are more complicated, often combining several steps or methods to encrypt data to keep it more secure and harder to break. Some modern methods make use of matrices as part of the encryption and decryption process; other fields of mathematics such as number theory play a large role in modern cryptography.

To use matrices in encoding and decoding secret messages, our procedure is as follows.

We first convert the secret message into a string of numbers by arbitrarily assigning a number to each letter of the message. Next we convert this string of numbers into a new set of numbers by multiplying the string by a square matrix of our choice that has an inverse. This new set of numbers represents the coded message.

To decode the message, we take the string of coded numbers and multiply it by the inverse of the matrix to get the original string of numbers. Finally, by associating the numbers with their corresponding letters, we obtain the original message.

In this section, we will use the correspondence shown below where letters A to Z correspond to the numbers 1 to 26, a space is represented by the number 27, and punctuation is ignored.

\[\begin{array}{cccccccccccccc} \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} & \mathrm{E} & \mathrm{F} & \mathrm{G} & \mathrm{H} & \mathrm{I} & \mathrm{J} & \mathrm{K} & \mathrm{L} & \mathrm{M} \\ 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\ \hline \mathrm{N} & \mathrm{O} & \mathrm{P} & \mathrm{Q} & \mathrm{R} & \mathrm{S} & \mathrm{T} & \mathrm{U} & \mathrm{V} & \mathrm{W} & \mathrm{X} & \mathrm{Y} & \mathrm{Z} \\ 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 & 26 \end{array} \nonumber \]

Example \(\PageIndex{1}\)

Use matrix \(A=\left[\begin{array}{ll} 1 & 2 \\ 1 & 3 \end{array}\right]\) to encode the message: ATTACK NOW!

We divide the letters of the message into groups of two.

\[\begin{array}{lllll}\text { AT } & \text { TA } & \text { CK } & \text { -N } & \text { OW }\end{array} \nonumber \]

We assign the numbers to these letters from the above table, and convert each pair of numbers into \(2 \times 1\) matrices. In the case where a single letter is left over on the end, a space is added to make it into a pair.

\[\left[\begin{array}{c} \mathrm{A} \\ \mathrm{T} \end{array}\right]=\left[\begin{array}{c} 1 \\ 20 \end{array}\right] \quad\left[\begin{array}{l} \mathrm{T} \\ \mathrm{A} \end{array}\right]=\left[\begin{array}{c} 20 \\ 1 \end{array}\right] \quad\left[\begin{array}{l} \mathrm{C} \\ \mathrm{K} \end{array}\right]=\left[\begin{array}{l} 3 \\ 11 \end{array}\right] \nonumber \]

\[\left[\begin{array}{c} - \\ \mathrm{N} \end{array}\right]=\left[\begin{array}{l} 27 \\ 14 \end{array}\right] \quad\left[\begin{array}{l} \mathrm{O} \\ \mathrm{W} \end{array}\right]=\left[\begin{array}{l} 15 \\ 23 \end{array}\right] \nonumber \]

So at this stage, our message expressed as \(2 \times 1\) matrices is as follows.

\[\left[\begin{array}{c} 1 \\ 20 \end{array}\right]\left[\begin{array}{c} 20 \\ 1 \end{array}\right]\left[\begin{array}{c} 3 \\ 11 \end{array}\right]\left[\begin{array}{c} 27 \\ 14 \end{array}\right]\left[\begin{array}{c} 15 \\ 23 \end{array}\right] \quad \bf{(I)} \nonumber \]

Now to encode, we multiply, on the left, each matrix of our message by the matrix \(A\). For example, the product of \(A\) with our first matrix is: \(\left[\begin{array}{ll} 1 & 2 \\ 1 & 3 \end{array}\right]\left[\begin{array}{c} 1 \\ 20 \end{array}\right]=\left[\begin{array}{l} 41 \\ 61 \end{array}\right]\)

And the product of \(A\) with our second matrix is: \(\left[\begin{array}{ll} 1 & 2 \\ 1 & 3 \end{array}\right]\left[\begin{array}{l} 20 \\ 1 \end{array}\right]=\left[\begin{array}{l} 22 \\ 23 \end{array}\right]\)

Multiplying each matrix in \(\bf{(I)}\) by matrix \(A\), in turn, gives the desired coded message:

\[\left[\begin{array}{l} 41 \\ 61 \end{array}\right]\left[\begin{array}{l} 22 \\ 23 \end{array}\right]\left[\begin{array}{l} 25 \\ 36 \end{array}\right]\left[\begin{array}{l} 55 \\ 69 \end{array}\right]\left[\begin{array}{l} 61 \\ 84 \end{array}\right] \nonumber \]

Example \(\PageIndex{2}\)

Decode the following message that was encoded using matrix \(A=\left[\begin{array}{ll} 1 & 2 \\ 1 & 3 \end{array}\right]\).

\[\left[\begin{array}{l} 21 \\ 26 \end{array}\right]\left[\begin{array}{l} 37 \\ 53 \end{array}\right]\left[\begin{array}{l} 45 \\ 54 \end{array}\right]\left[\begin{array}{c} 74 \\ 101 \end{array}\right]\left[\begin{array}{l} 53 \\ 69 \end{array}\right] \quad(\bf { II }) \nonumber \]

Since this message was encoded by multiplying by the matrix \(A\) in Example \(\PageIndex{1}\), we decode this message by first multiplying each matrix, on the left, by the inverse of matrix \(A\) given below.

\[A^{-1}=\left[\begin{array}{cc} 3 & -2 \\ -1 & 1 \end{array}\right] \nonumber \]

For example: \(\left[\begin{array}{cc} 3 & -2 \\ -1 & 1 \end{array}\right]\left[\begin{array}{c} 21 \\ 26 \end{array}\right]=\left[\begin{array}{c} 11 \\ 5 \end{array}\right]\)

By multiplying each of the matrices in \(\bf{(II)}\) by the matrix \(A^{-1}\), we get the following.

\[\left[\begin{array}{c} 11 \\ 5 \end{array}\right]\left[\begin{array}{c} 5 \\ 16 \end{array}\right]\left[\begin{array}{c} 27 \\ 9 \end{array}\right]\left[\begin{array}{c} 20 \\ 27 \end{array}\right]\left[\begin{array}{c} 21 \\ 16 \end{array}\right] \nonumber \]

Finally, by associating the numbers with their corresponding letters, we obtain:

\[\left[\begin{array}{l} \mathrm{K} \\ \mathrm{E} \end{array}\right]\left[\begin{array}{l} \mathrm{E} \\ \mathrm{P} \end{array}\right]\left[\begin{array}{l} - \\ \mathrm{I} \end{array}\right]\left[\begin{array}{l} \mathrm{T} \\ - \end{array}\right]\left[\begin{array}{l} \mathrm{U} \\ \mathrm{P} \end{array}\right] \nonumber \]

And the message reads: KEEP IT UP.

Now suppose we wanted to use a \(3 \times 3\) matrix to encode a message, then instead of dividing the letters into groups of two, we would divide them into groups of three.

Example \(\PageIndex{3}\)

Using the matrix \(B=\left[\begin{array}{ccc} 1 & 1 & -1 \\ 1 & 0 & 1 \\ 2 & 1 & 1 \end{array}\right] \nonumber\), encode the message: ATTACK NOW!

We divide the letters of the message into groups of three.

ATT ACK -NO W- -

Note that since the single letter "W" was left over on the end, we added two spaces to make it into a triplet.

Now we assign the numbers their corresponding letters from the table, and convert each triplet of numbers into \(3 \times 1\) matrices. We get

\[\left[\begin{array}{l} \mathrm{A} \\ \mathrm{T} \\ \mathrm{T} \end{array}\right]=\left[\begin{array}{c} 1 \\ 20 \\ 20 \end{array}\right]\left[\begin{array}{l} \mathrm{A} \\ \mathrm{C} \\ \mathrm{K} \end{array}\right]=\left[\begin{array}{l} 1 \\ 3 \\ 11 \end{array}\right]\left[\begin{array}{l} - \\ \mathrm{N} \\ \mathrm{O} \end{array}\right]=\left[\begin{array}{l} 27 \\ 14 \\ 15 \end{array}\right]\left[\begin{array}{l} \mathrm{W} \\ - \\ - \end{array}\right]=\left[\begin{array}{l} 23 \\ 27 \\ 27 \end{array}\right] \nonumber \]

So far we have,

\[\left[\begin{array}{c} 1 \\ 20 \\ 20 \end{array}\right]\left[\begin{array}{c} 1 \\ 3 \\ 11 \end{array}\right]\left[\begin{array}{c} 27 \\ 14 \\ 15 \end{array}\right]\left[\begin{array}{c} 23 \\ 27 \\ 27 \end{array}\right] \quad \bf{(III)} \nonumber \]

We multiply, on the left, each matrix of our message by the matrix \(B\). For example,

\[\left[\begin{array}{ccc} 1 & 1 & -1 \\ 1 & 0 & 1 \\ 2 & 1 & 1 \end{array}\right]\left[\begin{array}{c} 1 \\ 20 \\ 20 \end{array}\right]=\left[\begin{array}{c} 1 \\ 21 \\ 42 \end{array}\right] \nonumber \]

By multiplying each of the matrices in \(\bf{(III)}\) by the matrix \(B\), we get the desired coded message as follows:

\[\left[\begin{array}{c} 1 \\ 21 \\ 42 \end{array}\right]\left[\begin{array}{c} -7 \\ 12 \\ 16 \end{array}\right]\left[\begin{array}{c} 26 \\ 42 \\ 83 \end{array}\right]\left[\begin{array}{c} 23 \\ 50 \\ 100 \end{array}\right] \nonumber \]

If we need to decode this message, we simply multiply the coded message by \(B^{-1}\), and associate the numbers with the corresponding letters of the alphabet.

In Example \(\PageIndex{4}\) we will demonstrate how to use matrix \(B^{-1}\) to decode an encrypted message.

Example \(\PageIndex{4}\)

Decode the following message that was encoded using matrix \(B=\left[\begin{array}{lll} 1 & 1 & -1 \\ 1 & 0 & 1 \\ 2 & 1 & 1 \end{array}\right] \).

\[\left[\begin{array}{l} 11 \\ 20 \\ 43 \end{array}\right]\left[\begin{array}{l} 25 \\ 10 \\ 41 \end{array}\right]\left[\begin{array}{l} 22 \\ 14 \\ 41 \end{array}\right] \quad \bf{(IV)} \nonumber \]

Since this message was encoded by multiplying by the matrix \(B\). We first determine inverse of \(B\).

\[\mathrm{B}^{-1}=\left[\begin{array}{ccc} 1 & 2 & -1 \\ -1 & -3 & 2 \\ -1 & -1 & 1 \end{array}\right] \nonumber \]

To decode the message, we multiply each matrix, on the left, by \(B^{-1}\). For example,

\[\left[\begin{array}{ccc} 1 & 2 & -1 \\ -1 & -3 & 2 \\ -1 & -1 & 1 \end{array}\right]\left[\begin{array}{c} 11 \\ 20 \\ 43 \end{array}\right]=\left[\begin{array}{c} 8 \\ 15 \\ 12 \end{array}\right] \nonumber \]

Multiplying each of the matrices in \(\bf{(IV)}\) by the matrix \(B^{-1}\) gives the following.

\[\left[\begin{array}{c} 8 \\ 15 \\ 12 \end{array}\right]\left[\begin{array}{c} 4 \\ 27 \\ 6 \end{array}\right]\left[\begin{array}{c} 9 \\ 18 \\ 5 \end{array}\right] \nonumber \]

Finally, by associating the numbers with their corresponding letters, we obtain

\[\left[\begin{array}{l} \mathrm{H} \\ \mathrm{O} \\ \mathrm{L} \end{array}\right]\left[\begin{array}{l} \mathrm{D} \\ - \\ \mathrm{F} \end{array}\right]\left[\begin{array}{l} \mathrm{I} \\ \mathrm{R} \\ \mathrm{E} \end{array}\right] \quad \text { The message reads: } \mathrm{HOLD} \text { FIRE } \nonumber \]

The message reads: HOLD FIRE.

We summarize:

TO ENCODE A MESSAGE

1. Divide the letters of the message into groups of two or three.

2. Convert each group into a string of numbers by assigning a number to each letter of the message. Remember to assign letters to blank spaces.

3. Convert each group of numbers into column matrices.

3. Convert these column matrices into a new set of column matrices by multiplying them with a compatible square matrix of your choice that has an inverse. This new set of numbers or matrices represents the coded message.

TO DECODE A MESSAGE

1. Take the string of coded numbers and multiply it by the inverse of the matrix that was used to encode the message.

2. Associate the numbers with their corresponding letters.