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M5.1 Matrices and Matrix Operations

A matrix is an array of numbers arranged in rows and columns. Matrices, which are usually enclosed in parentheses or brackets, are used as an effective means of presenting or summarizing business data.

The following 2-row by 3-column $(2 \times 3)$ $(2 \times 3)$ matrix, for example, might be used by television station executives to describe the channel-switching behavior of their 5 o’clock TV news audience:

A table shows the probabilities of switching channels for the example.

5.1-1 Full Alternative Text

The number in the first row and first column indicates that there is a 0.80 probability that someone currently watching the Channel 6 news will continue to do so next month. Similarly, 15% of Channel 6’s viewers are expected to switch to Channel 8 next month (row 1, column 2), 5% will not be watching the 5 o’clock news at all (row 1, column 3), and so on for the second row.

The remainder of this module deals with the numerous mathematical operations that can be performed on matrices. These include adding, subtracting, and multiplying matrices; transposing a matrix; finding a matrix’s determinant, cofactors, and adjoint; and finding the inverse of a matrix.

Matrix Addition and Subtraction

Matrix addition and subtraction are the easiest operations. Matrices of the same dimensions—that is, the same number of rows and columns—can be added or subtracted by adding or subtracting the numbers in the same row and column of each matrix. Here are two small matrices:

\begin{matrix} Matrix A = (\begin{matrix} 5 & 7 \\ 2 & 1 \end{matrix}) \\ Matrix B = (\begin{matrix} 3 & 6 \\ 3 & 8 \end{matrix}) \end{matrix}

$\begin{matrix} Matrix A = (\begin{matrix} 5 & 7 \\ 2 & 1 \end{matrix}) \\ Matrix B = (\begin{matrix} 3 & 6 \\ 3 & 8 \end{matrix}) \end{matrix}$

To find the sum of these $2 \times 2$ $2 \times 2$ matrices, we add corresponding elements to create a new matrix:

An image shows the addition of two matrices.

5.1-2 Full Alternative Text

To subtract matrix B from matrix A, we simply subtract the corresponding elements in each position:

An image shows the subtraction of two matrices.

5.1-3 Full Alternative Text

Matrix Multiplication

Matrix multiplication is an operation that can take place only if the number of columns in the first matrix equals the number of rows in the second matrix. Thus, matrices of the dimensions in the following table can be multiplied:

MATRIX A SIZE	MATRIX B SIZE	SIZE OF $A × B$ $A × B$ RESULTING
$3 \times 3$ $3 \times 3$	$3 \times 3$ $3 \times 3$	$3 \times 3$ $3 \times 3$
$3 \times 1$ $3 \times 1$	$1 \times 3$ $1 \times 3$	$3 \times 3$ $3 \times 3$
$3 \times 1$ $3 \times 1$	$1 \times 1$ $1 \times 1$	$3 \times 1$ $3 \times 1$
$2 \times 4$ $2 \times 4$	$4 \times 3$ $4 \times 3$	$2 \times 3$ $2 \times 3$
$6 \times 9$ $6 \times 9$	$9 \times 2$ $9 \times 2$	$6 \times 2$ $6 \times 2$
$8 \times 3$ $8 \times 3$	$3 \times 6$ $3 \times 6$	$8 \times 6$ $8 \times 6$

We also note, in the far right column in the table, that the outer two numbers in the matrix sizes determine the dimensions of the new matrix. That is, if an 8-row by 3-column matrix is multiplied by a 3-row by 6-column matrix, the resultant product will be an 8-row by 6-column matrix.

Matrices of the dimensions in the following table may not be multiplied:

MATRIX A SIZE	MATRIX B SIZE
$3 \times 4$ $3 \times 4$	$3 \times 3$ $3 \times 3$
$1 \times 2$ $1 \times 2$	$1 \times 2$ $1 \times 2$
$6 \times 9$ $6 \times 9$	$8 \times 9$ $8 \times 9$
$2 \times 2$ $2 \times 2$	$3 \times 3$ $3 \times 3$

To perform the multiplication process, we take each row of the first matrix and multiply its elements times the numbers in each column of the second matrix. Hence, the number in the first row and first column of the new matrix is derived from the product of the first row of the first matrix times the first column of the second matrix. Similarly, the number in the first row and second column of the new matrix is the product of the first row of the first matrix times the second column of the second matrix. This concept is not nearly as confusing as it may sound.

Let us begin by computing the value of matrix C, which is the product of matrix A times matrix B:

Matrix A = {\begin{matrix} 5 \\ 2 \\ 3 \end{matrix}} Matrix B = (4 6)

$Matrix A = {\begin{matrix} 5 \\ 2 \\ 3 \end{matrix}} Matrix B = (4 6)$

This is a legitimate task, since matrix A is $3 \times 1$ $3 \times 1$ and matrix B is $1 \times 2.$ $1 \times 2.$ The product, matrix C, will have 3 rows and 2 columns $(3 \times 2) .$ $(3 \times 2) .$

Symbolically, the operation is $Matrix A \times Matrix B = Matrix C,$ $Matrix A \times Matrix B = Matrix C,$ or $A B = C :$ $A B = C :$

(\begin{matrix} a \\ b \\ c \end{matrix}) \times (d e) = (\begin{matrix} a d & a e \\ b d & b e \\ c d & c e \end{matrix})

$(\begin{matrix} a \\ b \\ c \end{matrix}) \times (d e) = (\begin{matrix} a d & a e \\ b d & b e \\ c d & c e \end{matrix})$ (M5-1)

Using the actual numbers, we have

(\begin{matrix} 5 \\ 2 \\ 3 \end{matrix}) \times (4 6) = (\begin{matrix} 20 & 30 \\ 8 & 12 \\ 12 & 18 \end{matrix}) = Matrix C

$(\begin{matrix} 5 \\ 2 \\ 3 \end{matrix}) \times (4 6) = (\begin{matrix} 20 & 30 \\ 8 & 12 \\ 12 & 18 \end{matrix}) = Matrix C$

As a second example, let matrix R be (6 2 5) and matrix S be

(\begin{matrix} 3 \\ 1 \\ 2 \end{matrix})

$(\begin{matrix} 3 \\ 1 \\ 2 \end{matrix})$

Then the product, $Matrix T = Matrix R \times Matrix S,$ $Matrix T = Matrix R \times Matrix S,$ will be of dimension $1 \times 1$ $1 \times 1$ because we are multiplying a $1 \times 3$ $1 \times 3$ matrix by a $3 \times 1$ $3 \times 1$ matrix:

\begin{array}{l} Matrix R \times Matrix S & = & Matrix T \\ (1 \times 3) (3 \times 1) & (1 \times 1) \\ (a b c) \times (\begin{array}{l} d \\ e \\ f \end{array}) & = & (a d + b e + c f) \\ (6 2 5) \times (\begin{array}{l} 3 \\ 1 \\ 2 \end{array}) & = & [(6) (3) + (2) (1) + (5) (2)] = (30) \end{array}

$\begin{array}{l} Matrix R \times Matrix S & = & Matrix T \\ (1 \times 3) (3 \times 1) & (1 \times 1) \\ (a b c) \times (\begin{array}{l} d \\ e \\ f \end{array}) & = & (a d + b e + c f) \\ (6 2 5) \times (\begin{array}{l} 3 \\ 1 \\ 2 \end{array}) & = & [(6) (3) + (2) (1) + (5) (2)] = (30) \end{array}$

To multiply larger matrices, we combine the approaches of the preceding examples:

\begin{array}{l} Matrix U = (\begin{array}{l} 6 & 2 \\ 7 & 1 \end{array}) & Matrix V = (\begin{array}{l} 3 & 4 \\ 5 & 8 \end{array}) \\ Matrix U \times Matrix V = & Matrix Y \\ (2 \times 2) \times (2 \times 2) & (2 \times 2) \\ (\begin{array}{l} a & b \\ c & d \end{array}) \times (\begin{array}{l} e & f \\ g & h \end{array}) & = & (\begin{array}{l} a e + b g & a f + b h \\ c e + d g & c f + d h \end{array}) \\ (\begin{array}{l} 6 & 2 \\ 7 & 1 \end{array}) \times (\begin{array}{l} 3 & 4 \\ 5 & 8 \end{array}) & = & (\begin{array}{l} 18 + 10 & 24 + 16 \\ 21 + 5 & 28 + 8 \end{array}) \\ = & (\begin{array}{l} 28 & 40 \\ 26 & 36 \end{array}) \end{array}

$\begin{array}{l} Matrix U = (\begin{array}{l} 6 & 2 \\ 7 & 1 \end{array}) & Matrix V = (\begin{array}{l} 3 & 4 \\ 5 & 8 \end{array}) \\ Matrix U \times Matrix V = & Matrix Y \\ (2 \times 2) \times (2 \times 2) & (2 \times 2) \\ (\begin{array}{l} a & b \\ c & d \end{array}) \times (\begin{array}{l} e & f \\ g & h \end{array}) & = & (\begin{array}{l} a e + b g & a f + b h \\ c e + d g & c f + d h \end{array}) \\ (\begin{array}{l} 6 & 2 \\ 7 & 1 \end{array}) \times (\begin{array}{l} 3 & 4 \\ 5 & 8 \end{array}) & = & (\begin{array}{l} 18 + 10 & 24 + 16 \\ 21 + 5 & 28 + 8 \end{array}) \\ = & (\begin{array}{l} 28 & 40 \\ 26 & 36 \end{array}) \end{array}$ (M5-2)

To introduce a special type of matrix, called the identity matrix, let’s try a final multiplication example:

\begin{array}{l} Matrix H & = & (\begin{array}{l} 4 & 7 \\ 2 & 3 \end{array}) & Matrix I & = (\begin{array}{l} 1 & 0 \\ 0 & 1 \end{array}) \\ Matrix H & \times & Matrix I & = & Matrix J \\ (\begin{array}{l} 4 & 7 \\ 2 & 3 \end{array}) & \times & (\begin{array}{l} 1 & 0 \\ 0 & 1 \end{array}) & = & (\begin{array}{l} 4 + 0 & 0 + 7 \\ 2 + 0 & 0 + 3 \end{array}) \\ = & (\begin{array}{l} 4 & 7 \\ 2 & 3 \end{array}) \end{array}

$\begin{array}{l} Matrix H & = & (\begin{array}{l} 4 & 7 \\ 2 & 3 \end{array}) & Matrix I & = (\begin{array}{l} 1 & 0 \\ 0 & 1 \end{array}) \\ Matrix H & \times & Matrix I & = & Matrix J \\ (\begin{array}{l} 4 & 7 \\ 2 & 3 \end{array}) & \times & (\begin{array}{l} 1 & 0 \\ 0 & 1 \end{array}) & = & (\begin{array}{l} 4 + 0 & 0 + 7 \\ 2 + 0 & 0 + 3 \end{array}) \\ = & (\begin{array}{l} 4 & 7 \\ 2 & 3 \end{array}) \end{array}$

Matrix I is called an identity matrix. An identity matrix has 1s on its primary diagonal and 0s in all other positions. When multiplied by any matrix of the same square dimensions, it yields the original matrix, so in this case, $Matrix J = Matrix H .$ $Matrix J = Matrix H .$

Matrix multiplication can also be useful in performing business computations. For example, Blank Plumbing and Heating is about to bid on three contract jobs: to install plumbing fixtures in a new university dormitory, an office building, and an apartment complex. The number of toilets, sinks, and bathtubs needed at each project is summarized in matrix notation as follows. The cost per plumbing fixture is also given. Matrix multiplication can be used to provide an estimate of the total cost of fixtures at each job.

Two tables show the number of plumbing fixtures to be installed and the cost per unit.

5.1-4 Full Alternative Text

Job demand $matrix \times Fixture cost matrix = Job$ $matrix \times Fixture cost matrix = Job$ cost matrix

\begin{array}{l} (3 \times 3) & (3 \times 1) & (3 \times 1) \\ (\begin{array}{l} 5 & 10 & 2 \\ 20 & 20 & 0 \\ 15 & 30 & 15 \end{array}) & \times & (\begin{array}{l} $ 40 \\ $ 25 \\ $ 50 \end{array}) & = & (\begin{array}{l} $ 200 & + & 250 & + & 100 \\ $ 800 & + & 500 & + & 0 \\ $ 600 & + & 750 & + & 750 \end{array}) & = & (\begin{array}{l} $ 550 \\ $ 1, 300 \\ $ 2, 100 \end{array}) \end{array}

$\begin{array}{l} (3 \times 3) & (3 \times 1) & (3 \times 1) \\ (\begin{array}{l} 5 & 10 & 2 \\ 20 & 20 & 0 \\ 15 & 30 & 15 \end{array}) & \times & (\begin{array}{l} $ 40 \\ $ 25 \\ $ 50 \end{array}) & = & (\begin{array}{l} $ 200 & + & 250 & + & 100 \\ $ 800 & + & 500 & + & 0 \\ $ 600 & + & 750 & + & 750 \end{array}) & = & (\begin{array}{l} $ 550 \\ $ 1, 300 \\ $ 2, 100 \end{array}) \end{array}$

Hence, Blank Plumbing can expect to spend $550 on fixtures at the dormitory project, $1,300 at the office building, and $2,100 at the apartment complex.

Matrix Notation for Systems of Equations

The use of matrices is helpful in representing a system of equations. For example, the system

\begin{array}{l} 2 X_{1} & + & 3 X_{2} & = & 24 \\ 4 X_{1} & + & 2 X_{2} & = & 36 \end{array}

$\begin{array}{l} 2 X_{1} & + & 3 X_{2} & = & 24 \\ 4 X_{1} & + & 2 X_{2} & = & 36 \end{array}$

can be written as

(\begin{matrix} 2 & 3 \\ 4 & 2 \end{matrix}) (\begin{matrix} X_{1} \\ X_{2} \end{matrix}) = (\begin{matrix} 24 \\ 36 \end{matrix})

$(\begin{matrix} 2 & 3 \\ 4 & 2 \end{matrix}) (\begin{matrix} X_{1} \\ X_{2} \end{matrix}) = (\begin{matrix} 24 \\ 36 \end{matrix})$

In general, we can express a system of equations as

A X = B

$A X = B$

Matrix Transpose

The transpose of a matrix is a means of presenting data in a different form. To create the transpose of a given matrix, we simply interchange the rows with the columns. Hence, the first row of a matrix becomes its first column, the second row becomes its second column, and so on.

Two matrices are transposed here:

\begin{array}{l} Matrix A & = & (\begin{array}{l} 5 & 2 & 6 \\ 3 & 0 & 9 \\ 1 & 4 & 8 \end{array}) \\ Transpose of Matrix A & = & (\begin{array}{l} 5 & 3 & 1 \\ 2 & 0 & 4 \\ 6 & 9 & 8 \end{array}) \\ Matrix B & = & (\begin{array}{l} 2 & 7 & 0 & 3 \\ 8 & 5 & 6 & 4 \end{array}) \\ Transpose of Matrix B & = & (\begin{array}{l} 2 & 8 \\ 7 & 5 \\ 0 & 6 \\ 3 & 4 \end{array}) \end{array}

$\begin{array}{l} Matrix A & = & (\begin{array}{l} 5 & 2 & 6 \\ 3 & 0 & 9 \\ 1 & 4 & 8 \end{array}) \\ Transpose of Matrix A & = & (\begin{array}{l} 5 & 3 & 1 \\ 2 & 0 & 4 \\ 6 & 9 & 8 \end{array}) \\ Matrix B & = & (\begin{array}{l} 2 & 7 & 0 & 3 \\ 8 & 5 & 6 & 4 \end{array}) \\ Transpose of Matrix B & = & (\begin{array}{l} 2 & 8 \\ 7 & 5 \\ 0 & 6 \\ 3 & 4 \end{array}) \end{array}$

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Table of Contents for M5.1 Matrices and Matrix Operations

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Table of Contents for
M5.1 Matrices and Matrix Operations