Matrix Subtraction

Subtracting Entries in Matching Positions

Matrix subtraction is an operation to find the difference between two matrices. Just like matrix addition, the subtraction operation can only be performed if both matrices involved have the same size or order.

The result of matrix subtraction is a new matrix that also has the same order, where each element is the result of subtracting the corresponding elements from the two initial matrices.

Formal Definition of Matrix Subtraction

There are two common ways to define matrix subtraction, both of which lead to the same result.

Subtraction as Addition with the Opposite

The subtraction of matrix $B$ from matrix $A$ can be defined as the addition of matrix $A$ with the opposite matrix of $B$ (i.e., $-B$ ).

A - B = A + (-B)

Matrix $-B$ is the matrix obtained by multiplying each element of matrix $B$ by $-1$ . So, if $B = [b_{ij}]$ , then $-B = [-b_{ij}]$ .

Subtraction of Corresponding Elements

If matrix $A = [a_{ij}]$ and matrix $B = [b_{ij}]$ both have the order $m \times n$ , then the result of the subtraction $C = A - B$ will also be of order $m \times n$ .

Each element $c_{ij}$ of matrix $C$ is calculated by subtracting the corresponding element of matrix $B$ from the corresponding element of matrix $A$ :

c_{ij} = a_{ij} - b_{ij}

This means we subtract the elements that are in the same row and column position.

Both definitions are equivalent and will produce the same difference matrix.

How to Perform Matrix Subtraction

To subtract two matrices, follow these steps:

Ensure Same Order: Check that both matrices have the same number of rows and columns. If their orders differ, subtraction is undefined.
Subtract Corresponding Elements: If the orders are the same, subtract each element of the second matrix (subtrahend) from the corresponding element of the first matrix.
Form the Resultant Matrix: Arrange the results of these subtractions into a new matrix. This new matrix will have the same order as the initial matrices.

Example of Matrix Subtraction

Suppose we have two matrices, $P$ and $Q$ , as follows:

P = \begin{bmatrix} 8 & 5 \\ 3 & 7 \end{bmatrix}

Both matrices are of order $2 \times 2$ , so they can be subtracted.

Using the method of subtracting corresponding elements:

P - Q = \begin{bmatrix} 8 & 5 \\ 3 & 7 \end{bmatrix} - \begin{bmatrix} 2 & 1 \\ -1 & 4 \end{bmatrix}

Using the method of addition with the opposite ( $P + (-Q)$ ):

First, determine $-Q$ :

-Q = \begin{bmatrix} -2 & -1 \\ -(-1) & -4 \end{bmatrix} = \begin{bmatrix} -2 & -1 \\ 1 & -4 \end{bmatrix}

Then, add $P$ to $-Q$ :

P + (-Q) = \begin{bmatrix} 8 & 5 \\ 3 & 7 \end{bmatrix} + \begin{bmatrix} -2 & -1 \\ 1 & -4 \end{bmatrix}

Both methods yield the same matrix.

Example of Matrices That Cannot Be Subtracted

These two matrices do not have the same order:

K = \begin{bmatrix} 1 & 0 \\ -2 & 5 \\ 3 & 1 \end{bmatrix}

Matrix $K$ is of order $3 \times 2$ , while matrix $L$ is of order $2 \times 2$ . Since the orders of these two matrices are different, the subtraction $K-L$ (or $L-K$ ) cannot be performed or is undefined.

Properties of Matrix Subtraction

Matrix addition is commutative and associative, but matrix subtraction generally has neither property.

Not Commutative: In general, the order of matrix subtraction significantly affects the result. This means $A - B$ is not equal to $B - A$ , except in special cases (e.g., if $A=B$ ).

In general:

$A - B \neq B - A$

For example, from matrices $P$ and $Q$ above:

$P-Q = \begin{bmatrix} 6 & 4 \\ 4 & 3 \end{bmatrix}$

Whereas,
$Q-P = \begin{bmatrix} 2 & 1 \\ -1 & 4 \end{bmatrix} - \begin{bmatrix} 8 & 5 \\ 3 & 7 \end{bmatrix}$
Therefore, $P-Q \neq Q-P$ .
Not Associative: The grouping in the subtraction of three or more matrices also affects the final result. In general, $(A - B) - C$ is not equal to $A - (B - C)$ .

In general:

$(A - B) - C \neq A - (B - C)$

This is because $(A - B) - C = A - B - C$ , whereas $A - (B - C) = A - B + C$ .

Matrix subtraction can be written as addition of the opposite matrix: $A - B = A + (-B)$ .

By converting subtraction into addition with the opposite matrix, we can use addition properties when needed.

Exercises

Problem 1

Given the following matrices:

M = \begin{bmatrix} 4 & -2 & 0 \\ 1 & 5 & 3 \end{bmatrix}

Determine the result of $M-N$ .

Problem 2

Determine the values of $x, y,$ and $z$ from the following matrix equation:

\begin{aligned} &\begin{bmatrix} 2x & 7 \\ y-1 & 5 \end{bmatrix} - \begin{bmatrix} 3 & z \\ 4 & -2x \end{bmatrix} \\ &= \begin{bmatrix} 5 & 10 \\ -3 & 13 \end{bmatrix} \end{aligned}

Problem 3

Given three matrices:

A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

Calculate $(A-B)-C$ and $A-(B-C)$ . Are the results the same?

Worked Solutions

Problem 1

Given:

M = \begin{bmatrix} 4 & -2 & 0 \\ 1 & 5 & 3 \end{bmatrix}

Then, $M-N$ is:

\begin{aligned} M -N &= \begin{bmatrix} 4 & -2 & 0 \\ 1 & 5 & 3 \end{bmatrix} \\ &\quad{}- \begin{bmatrix} -1 & 1 & -2 \\ 6 & 0 & 7 \end{bmatrix} \end{aligned}

Problem 2

Given the matrix equation:

\begin{aligned} &\begin{bmatrix} 2x & 7 \\ y-1 & 5 \end{bmatrix} - \begin{bmatrix} 3 & z \\ 4 & -2x \end{bmatrix} \\ &= \begin{bmatrix} 5 & 10 \\ -3 & 13 \end{bmatrix} \end{aligned}

Perform the subtraction operation on the left side:

\begin{aligned} &\begin{bmatrix} 2x-3 & 7-z \\ (y-1)-4 & 5-(-2x) \end{bmatrix} \\ &= \begin{bmatrix} 5 & 10 \\ -3 & 13 \end{bmatrix} \end{aligned}

Based on the equality of two matrices, corresponding elements must be equal:

From the element in row $1$ , column $1$ : $2x-3 = 5$

2x = 5+3

From the element in row $1$ , column $2$ : $7-z = 10$

-z = 10-7

From the element in row $2$ , column $1$ : $y-5 = -3$

y = -3+5

From the element in row $2$ , column $2$ : $5+2x = 13$ .

Substituting $x=4$ from the first equation gives $5+2(4) = 13$ .

The shared bottom-right value $13$ confirms the value of $x$ . It adds no new unknown to solve.

The solution is $x=4$ , $y=2$ , and $z=-3$ .

Substituting these values into the original equation verifies all four corresponding entries.

Problem 3

Given:

A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

Calculate $(A-B)-C$ :

First, $A-B$ :

A-B = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} - \begin{bmatrix} 3 & -2 \\ 5 & 1 \end{bmatrix}

Then, $(A-B)-C$ :

\begin{aligned} (A-B) -C &= \begin{bmatrix} -2 & 2 \\ -5 & 0 \end{bmatrix} \\ &\quad{}- \begin{bmatrix} -4 & 0 \\ 2 & -3 \end{bmatrix} \end{aligned}

Calculate $A-(B-C)$ :

First, $B-C$ :

B-C = \begin{bmatrix} 3 & -2 \\ 5 & 1 \end{bmatrix} - \begin{bmatrix} -4 & 0 \\ 2 & -3 \end{bmatrix}

Then, $A-(B-C)$ :

A-(B-C) = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} - \begin{bmatrix} 7 & -2 \\ 3 & 4 \end{bmatrix}

The two results differ: $(A-B)-C \neq A-(B-C)$ .

Matrix subtraction is therefore not associative.

Subtracting Entries in Matching Positions

The result of matrix subtraction is a new matrix that also has the same order, where each element is the result of subtracting the corresponding elements from the two initial matrices.

Formal Definition of Matrix Subtraction

There are two common ways to define matrix subtraction, both of which lead to the same result.

Subtraction as Addition with the Opposite

The subtraction of matrix $B$ from matrix $A$ can be defined as the addition of matrix $A$ with the opposite matrix of $B$ (i.e., $-B$ ).

A - B = A + (-B)

Matrix $-B$ is the matrix obtained by multiplying each element of matrix $B$ by $-1$ . So, if $B = [b_{ij}]$ , then $-B = [-b_{ij}]$ .

Subtraction of Corresponding Elements

If matrix $A = [a_{ij}]$ and matrix $B = [b_{ij}]$ both have the order $m \times n$ , then the result of the subtraction $C = A - B$ will also be of order $m \times n$ .

Each element $c_{ij}$ of matrix $C$ is calculated by subtracting the corresponding element of matrix $B$ from the corresponding element of matrix $A$ :

c_{ij} = a_{ij} - b_{ij}

This means we subtract the elements that are in the same row and column position.

Both definitions are equivalent and will produce the same difference matrix.

How to Perform Matrix Subtraction

To subtract two matrices, follow these steps:

Ensure Same Order: Check that both matrices have the same number of rows and columns. If their orders differ, subtraction is undefined.
Subtract Corresponding Elements: If the orders are the same, subtract each element of the second matrix (subtrahend) from the corresponding element of the first matrix.
Form the Resultant Matrix: Arrange the results of these subtractions into a new matrix. This new matrix will have the same order as the initial matrices.

Example of Matrix Subtraction

Suppose we have two matrices, $P$ and $Q$ , as follows:

P = \begin{bmatrix} 8 & 5 \\ 3 & 7 \end{bmatrix}

Both matrices are of order $2 \times 2$ , so they can be subtracted.

Using the method of subtracting corresponding elements:

P - Q = \begin{bmatrix} 8 & 5 \\ 3 & 7 \end{bmatrix} - \begin{bmatrix} 2 & 1 \\ -1 & 4 \end{bmatrix}

Using the method of addition with the opposite ( $P + (-Q)$ ):

First, determine $-Q$ :

-Q = \begin{bmatrix} -2 & -1 \\ -(-1) & -4 \end{bmatrix} = \begin{bmatrix} -2 & -1 \\ 1 & -4 \end{bmatrix}

Then, add $P$ to $-Q$ :

P + (-Q) = \begin{bmatrix} 8 & 5 \\ 3 & 7 \end{bmatrix} + \begin{bmatrix} -2 & -1 \\ 1 & -4 \end{bmatrix}

Both methods yield the same matrix.

Example of Matrices That Cannot Be Subtracted

These two matrices do not have the same order:

K = \begin{bmatrix} 1 & 0 \\ -2 & 5 \\ 3 & 1 \end{bmatrix}

Properties of Matrix Subtraction

Matrix addition is commutative and associative, but matrix subtraction generally has neither property.

Not Commutative: In general, the order of matrix subtraction significantly affects the result. This means $A - B$ is not equal to $B - A$ , except in special cases (e.g., if $A=B$ ).

In general:

$A - B \neq B - A$

For example, from matrices $P$ and $Q$ above:

$P-Q = \begin{bmatrix} 6 & 4 \\ 4 & 3 \end{bmatrix}$

Whereas,
$Q-P = \begin{bmatrix} 2 & 1 \\ -1 & 4 \end{bmatrix} - \begin{bmatrix} 8 & 5 \\ 3 & 7 \end{bmatrix}$
Therefore, $P-Q \neq Q-P$ .
Not Associative: The grouping in the subtraction of three or more matrices also affects the final result. In general, $(A - B) - C$ is not equal to $A - (B - C)$ .

In general:

$(A - B) - C \neq A - (B - C)$

This is because $(A - B) - C = A - B - C$ , whereas $A - (B - C) = A - B + C$ .

Matrix subtraction can be written as addition of the opposite matrix: $A - B = A + (-B)$ .

By converting subtraction into addition with the opposite matrix, we can use addition properties when needed.

Exercises

Problem 1

Given the following matrices:

M = \begin{bmatrix} 4 & -2 & 0 \\ 1 & 5 & 3 \end{bmatrix}

Determine the result of $M-N$ .

Problem 2

Determine the values of $x, y,$ and $z$ from the following matrix equation:

\begin{aligned} &\begin{bmatrix} 2x & 7 \\ y-1 & 5 \end{bmatrix} - \begin{bmatrix} 3 & z \\ 4 & -2x \end{bmatrix} \\ &= \begin{bmatrix} 5 & 10 \\ -3 & 13 \end{bmatrix} \end{aligned}

Problem 3

Given three matrices:

A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

Calculate $(A-B)-C$ and $A-(B-C)$ . Are the results the same?

Worked Solutions

Problem 1

Given:

M = \begin{bmatrix} 4 & -2 & 0 \\ 1 & 5 & 3 \end{bmatrix}

Then, $M-N$ is:

\begin{aligned} M -N &= \begin{bmatrix} 4 & -2 & 0 \\ 1 & 5 & 3 \end{bmatrix} \\ &\quad{}- \begin{bmatrix} -1 & 1 & -2 \\ 6 & 0 & 7 \end{bmatrix} \end{aligned}

Problem 2

Given the matrix equation:

\begin{aligned} &\begin{bmatrix} 2x & 7 \\ y-1 & 5 \end{bmatrix} - \begin{bmatrix} 3 & z \\ 4 & -2x \end{bmatrix} \\ &= \begin{bmatrix} 5 & 10 \\ -3 & 13 \end{bmatrix} \end{aligned}

Perform the subtraction operation on the left side:

\begin{aligned} &\begin{bmatrix} 2x-3 & 7-z \\ (y-1)-4 & 5-(-2x) \end{bmatrix} \\ &= \begin{bmatrix} 5 & 10 \\ -3 & 13 \end{bmatrix} \end{aligned}

Based on the equality of two matrices, corresponding elements must be equal:

From the element in row $1$ , column $1$ : $2x-3 = 5$

2x = 5+3

From the element in row $1$ , column $2$ : $7-z = 10$

-z = 10-7

From the element in row $2$ , column $1$ : $y-5 = -3$

y = -3+5

From the element in row $2$ , column $2$ : $5+2x = 13$ .

Substituting $x=4$ from the first equation gives $5+2(4) = 13$ .

The shared bottom-right value $13$ confirms the value of $x$ . It adds no new unknown to solve.

The solution is $x=4$ , $y=2$ , and $z=-3$ .

Substituting these values into the original equation verifies all four corresponding entries.

Problem 3

Given:

A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

Calculate $(A-B)-C$ :

First, $A-B$ :

A-B = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} - \begin{bmatrix} 3 & -2 \\ 5 & 1 \end{bmatrix}

Then, $(A-B)-C$ :

\begin{aligned} (A-B) -C &= \begin{bmatrix} -2 & 2 \\ -5 & 0 \end{bmatrix} \\ &\quad{}- \begin{bmatrix} -4 & 0 \\ 2 & -3 \end{bmatrix} \end{aligned}

Calculate $A-(B-C)$ :

First, $B-C$ :

B-C = \begin{bmatrix} 3 & -2 \\ 5 & 1 \end{bmatrix} - \begin{bmatrix} -4 & 0 \\ 2 & -3 \end{bmatrix}

Then, $A-(B-C)$ :

A-(B-C) = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} - \begin{bmatrix} 7 & -2 \\ 3 & 4 \end{bmatrix}

The two results differ: $(A-B)-C \neq A-(B-C)$ .

Matrix subtraction is therefore not associative.