Matrix Addition

Adding Entries in Matching Positions

Matrix addition combines entries in the same positions. It is defined only for matrices with the same number of rows and columns.

The result of the addition is a new matrix of the same order, where each element is the sum of the corresponding elements (elements in the same position) from the original matrices.

Formal Definition of Matrix Addition

Two matrices, say matrix $A$ and matrix $B$ , can be added if and only if both matrices have the same order.

Suppose matrix $A$ is of order $m \times n$ with elements $a_{ij}$ (element in the $i$ -th row and $j$ -th column), and matrix $B$ is also of order $m \times n$ with elements $b_{ij}$ .

Call their sum matrix $C$ and write $C = A + B$ . Matrix $C$ also has dimensions $m \times n$ , with entries $c_{ij}$ defined by:

c_{ij} = a_{ij} + b_{ij}

For each position, the formula adds the entry from $A$ to the corresponding entry from $B$ .

In symbols, $A\in\mathbb{R}^{m\times n}$ and $B\in\mathbb{R}^{m\times n}$ imply $A+B\in\mathbb{R}^{m\times n}$ . Every output entry $c_{ij}$ therefore has one corresponding input entry in each matrix.

How to Perform Matrix Addition

To add two matrices, follow these steps:

Ensure Same Order: Check if both matrices have the same number of rows and columns. If not, addition cannot be performed.
Add Corresponding Elements: Add the elements that are in the same row and column position from both matrices.
Form the Resultant Matrix: Arrange the sums of these elements into a new matrix of the same order.

Example of Matrix Addition

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

P = \begin{bmatrix} 1 & 5 \\ -2 & 0 \\ 4 & 7 \end{bmatrix}

Both matrices are of order $3 \times 2$ ( $3$ rows and $2$ columns), so they can be added.

Then, $P+Q$ is:

P+Q = \begin{bmatrix} 1 & 5 \\ -2 & 0 \\ 4 & 7 \end{bmatrix} + \begin{bmatrix} 6 & -3 \\ 8 & 1 \\ -2 & 9 \end{bmatrix}

The matrix above is the sum $P+Q$ .

Matrices That Cannot Be Added

These two matrices have different orders:

X = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}

Matrix $X$ has dimensions $2 \times 2$ , while matrix $Y$ has dimensions $2 \times 3$ . Their column counts differ, so $X+Y$ is not defined.

Properties of Matrix Addition

Matrix addition follows the same basic laws as addition of real numbers. Let $A$ , $B$ , and $C$ have the same dimensions, and let $O$ be the zero matrix of those dimensions.

Commutative Property: The order of matrix addition does not affect the result.

$A + B = B + A$

Adding matrix $A$ to $B$ produces the same matrix as adding matrix $B$ to $A$ .
Associative Property: The grouping in the addition of three or more matrices does not affect the result.

$(A + B) + C = A + (B + C)$

You can add $A$ and $B$ first, then add the result to $C$ , or add $B$ and $C$ first, then add $A$ to the result. The final outcome will be the same.
Existence of an Identity Element (Zero Matrix): The zero matrix $O$ is the identity element for addition.

$A + O = O + A = A$

Adding the zero matrix leaves $A$ unchanged.

This zero matrix plays a role similar to the number $0$ in the addition of numbers.
Existence of an Additive Inverse (Opposite of a Matrix): Every matrix $A$ has an additive inverse, denoted as $-A$ , which when added to $A$ results in the zero matrix $O$ .

$A + (-A) = O$

The matrix $-A$ is a matrix where each element is the opposite (negative) of the corresponding elements of matrix $A$ .

For example, if $a_{ij}$ is an element of $A$ , then $-a_{ij}$ is an element of $-A$ .

Exercises

Problem 1

Given the following matrices:

A = \begin{bmatrix} 3 & -1 & 7 \\ 0 & 5 & 2 \end{bmatrix}

Calculate $A+B$ and $B+A$ . Then, determine if $A+C$ can be calculated and provide your explanation.

Problem 2

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

\begin{bmatrix} 2x & 5 \\ -1 & 3y \end{bmatrix} + \begin{bmatrix} 4 & -2 \\ z & 7 \end{bmatrix} = \begin{bmatrix} 10 & 3 \\ 6 & 1 \end{bmatrix}

Problem 3

For the matrix below, determine $-P$ (the additive inverse of $P$ ) and prove that $P + (-P) = O$ . Here, $O$ is the zero matrix of the same order.

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

Worked Solutions

Problem 1

Given matrices:

A = \begin{bmatrix} 3 & -1 & 7 \\ 0 & 5 & 2 \end{bmatrix}

Addition of matrix $A$ and $B$ ( $A+B$ ):

\begin{aligned} A +B &= \begin{bmatrix} 3 & -1 & 7 \\ 0 & 5 & 2 \end{bmatrix} \\ &\quad{}+ \begin{bmatrix} -2 & 9 & -4 \\ 6 & -3 & 1 \end{bmatrix} \end{aligned}

Adding a negative entry subtracts its opposite, so the corresponding sums become:

=\begin{bmatrix}3-2&-1+9&7-4\\0+6&5-3&2+1\end{bmatrix}

Addition of matrix $B$ and $A$ ( $B+A$ ):

\begin{aligned} B +A &= \begin{bmatrix} -2 & 9 & -4 \\ 6 & -3 & 1 \end{bmatrix} \\ &\quad{}+ \begin{bmatrix} 3 & -1 & 7 \\ 0 & 5 & 2 \end{bmatrix} \end{aligned}

(Commutative property proven: $A+B = B+A$ )

Addition of matrix $A$ and $C$ ( $A+C$ ):

Cannot be calculated. Matrix $A$ is of order $2 \times 3$ , while matrix $C$ is of order $2 \times 2$ . Since their orders are different, the addition $A+C$ cannot be performed.

Problem 2

Given the matrix addition:

\begin{bmatrix} 2x & 5 \\ -1 & 3y \end{bmatrix} + \begin{bmatrix} 4 & -2 \\ z & 7 \end{bmatrix} = \begin{bmatrix} 10 & 3 \\ 6 & 1 \end{bmatrix}

Perform matrix addition on the left side:

\begin{bmatrix} 2x+4 & 5+(-2) \\ -1+z & 3y+7 \end{bmatrix} = \begin{bmatrix} 10 & 3 \\ 6 & 1 \end{bmatrix}

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

For the element in row $1$ , column $1$ : $2x+4 = 10$

2x = 10-4

For the element in row $1$ , column $2$ : $3 = 3$ (already consistent).

For the element in row $2$ , column $1$ : $-1+z = 6$

z = 6+1

For the element in row $2$ , column $2$ : $3y+7 = 1$

3y = 1-7

The values are $x=3$ , $y=-2$ , and $z=7$ .

Problem 3

The given matrix is:

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

The additive inverse of $P$ , which is $-P$ , is:

-P = \begin{bmatrix} -1 & -(-2) \\ -3 & -0 \end{bmatrix} = \begin{bmatrix} -1 & 2 \\ -3 & 0 \end{bmatrix}

Proof that $P + (-P) = O$ :

P + (-P) = \begin{bmatrix} 1 & -2 \\ 3 & 0 \end{bmatrix} + \begin{bmatrix} -1 & 2 \\ -3 & 0 \end{bmatrix}

The result is the zero matrix $O$ of order $2 \times 2$ . Proven.

Adding Entries in Matching Positions

Matrix addition combines entries in the same positions. It is defined only for matrices with the same number of rows and columns.

The result of the addition is a new matrix of the same order, where each element is the sum of the corresponding elements (elements in the same position) from the original matrices.

Formal Definition of Matrix Addition

Two matrices, say matrix $A$ and matrix $B$ , can be added if and only if both matrices have the same order.

Suppose matrix $A$ is of order $m \times n$ with elements $a_{ij}$ (element in the $i$ -th row and $j$ -th column), and matrix $B$ is also of order $m \times n$ with elements $b_{ij}$ .

Call their sum matrix $C$ and write $C = A + B$ . Matrix $C$ also has dimensions $m \times n$ , with entries $c_{ij}$ defined by:

c_{ij} = a_{ij} + b_{ij}

For each position, the formula adds the entry from $A$ to the corresponding entry from $B$ .

How to Perform Matrix Addition

To add two matrices, follow these steps:

Ensure Same Order: Check if both matrices have the same number of rows and columns. If not, addition cannot be performed.
Add Corresponding Elements: Add the elements that are in the same row and column position from both matrices.
Form the Resultant Matrix: Arrange the sums of these elements into a new matrix of the same order.

Example of Matrix Addition

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

P = \begin{bmatrix} 1 & 5 \\ -2 & 0 \\ 4 & 7 \end{bmatrix}

Both matrices are of order $3 \times 2$ ( $3$ rows and $2$ columns), so they can be added.

Then, $P+Q$ is:

P+Q = \begin{bmatrix} 1 & 5 \\ -2 & 0 \\ 4 & 7 \end{bmatrix} + \begin{bmatrix} 6 & -3 \\ 8 & 1 \\ -2 & 9 \end{bmatrix}

The matrix above is the sum $P+Q$ .

Matrices That Cannot Be Added

These two matrices have different orders:

X = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}

Matrix $X$ has dimensions $2 \times 2$ , while matrix $Y$ has dimensions $2 \times 3$ . Their column counts differ, so $X+Y$ is not defined.

Properties of Matrix Addition

Matrix addition follows the same basic laws as addition of real numbers. Let $A$ , $B$ , and $C$ have the same dimensions, and let $O$ be the zero matrix of those dimensions.

Commutative Property: The order of matrix addition does not affect the result.

$A + B = B + A$

Adding matrix $A$ to $B$ produces the same matrix as adding matrix $B$ to $A$ .
Associative Property: The grouping in the addition of three or more matrices does not affect the result.

$(A + B) + C = A + (B + C)$

You can add $A$ and $B$ first, then add the result to $C$ , or add $B$ and $C$ first, then add $A$ to the result. The final outcome will be the same.
Existence of an Identity Element (Zero Matrix): The zero matrix $O$ is the identity element for addition.

$A + O = O + A = A$

Adding the zero matrix leaves $A$ unchanged.

This zero matrix plays a role similar to the number $0$ in the addition of numbers.
Existence of an Additive Inverse (Opposite of a Matrix): Every matrix $A$ has an additive inverse, denoted as $-A$ , which when added to $A$ results in the zero matrix $O$ .

$A + (-A) = O$

The matrix $-A$ is a matrix where each element is the opposite (negative) of the corresponding elements of matrix $A$ .

For example, if $a_{ij}$ is an element of $A$ , then $-a_{ij}$ is an element of $-A$ .

Exercises

Problem 1

Given the following matrices:

A = \begin{bmatrix} 3 & -1 & 7 \\ 0 & 5 & 2 \end{bmatrix}

Calculate $A+B$ and $B+A$ . Then, determine if $A+C$ can be calculated and provide your explanation.

Problem 2

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

\begin{bmatrix} 2x & 5 \\ -1 & 3y \end{bmatrix} + \begin{bmatrix} 4 & -2 \\ z & 7 \end{bmatrix} = \begin{bmatrix} 10 & 3 \\ 6 & 1 \end{bmatrix}

Problem 3

For the matrix below, determine $-P$ (the additive inverse of $P$ ) and prove that $P + (-P) = O$ . Here, $O$ is the zero matrix of the same order.

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

Worked Solutions

Problem 1

Given matrices:

A = \begin{bmatrix} 3 & -1 & 7 \\ 0 & 5 & 2 \end{bmatrix}

Addition of matrix $A$ and $B$ ( $A+B$ ):

\begin{aligned} A +B &= \begin{bmatrix} 3 & -1 & 7 \\ 0 & 5 & 2 \end{bmatrix} \\ &\quad{}+ \begin{bmatrix} -2 & 9 & -4 \\ 6 & -3 & 1 \end{bmatrix} \end{aligned}

Adding a negative entry subtracts its opposite, so the corresponding sums become:

=\begin{bmatrix}3-2&-1+9&7-4\\0+6&5-3&2+1\end{bmatrix}

Addition of matrix $B$ and $A$ ( $B+A$ ):

\begin{aligned} B +A &= \begin{bmatrix} -2 & 9 & -4 \\ 6 & -3 & 1 \end{bmatrix} \\ &\quad{}+ \begin{bmatrix} 3 & -1 & 7 \\ 0 & 5 & 2 \end{bmatrix} \end{aligned}

(Commutative property proven: $A+B = B+A$ )

Addition of matrix $A$ and $C$ ( $A+C$ ):

Cannot be calculated. Matrix $A$ is of order $2 \times 3$ , while matrix $C$ is of order $2 \times 2$ . Since their orders are different, the addition $A+C$ cannot be performed.

Problem 2

Given the matrix addition:

\begin{bmatrix} 2x & 5 \\ -1 & 3y \end{bmatrix} + \begin{bmatrix} 4 & -2 \\ z & 7 \end{bmatrix} = \begin{bmatrix} 10 & 3 \\ 6 & 1 \end{bmatrix}

Perform matrix addition on the left side:

\begin{bmatrix} 2x+4 & 5+(-2) \\ -1+z & 3y+7 \end{bmatrix} = \begin{bmatrix} 10 & 3 \\ 6 & 1 \end{bmatrix}

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

For the element in row $1$ , column $1$ : $2x+4 = 10$

2x = 10-4

For the element in row $1$ , column $2$ : $3 = 3$ (already consistent).

For the element in row $2$ , column $1$ : $-1+z = 6$

z = 6+1

For the element in row $2$ , column $2$ : $3y+7 = 1$

3y = 1-7

The values are $x=3$ , $y=-2$ , and $z=7$ .

Problem 3

The given matrix is:

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

The additive inverse of $P$ , which is $-P$ , is:

-P = \begin{bmatrix} -1 & -(-2) \\ -3 & -0 \end{bmatrix} = \begin{bmatrix} -1 & 2 \\ -3 & 0 \end{bmatrix}

Proof that $P + (-P) = O$ :

P + (-P) = \begin{bmatrix} 1 & -2 \\ 3 & 0 \end{bmatrix} + \begin{bmatrix} -1 & 2 \\ -3 & 0 \end{bmatrix}

The result is the zero matrix $O$ of order $2 \times 2$ . Proven.