Operations of Matrices: Addition, Subtraction and Properties

This lesson introduces the basic operations on matrices: addition, subtraction, multiplication by a scalar, and the main properties of matrix addition.

Adding and subtracting matrices

To add or subtract two matrices they must have the same order. You then add or subtract the corresponding elements, and the answer keeps that same order.

Worked example: addition

Given the two $3 \times 3$ matrices

$A = \begin{bmatrix} 1 & 3 & 4 \\ 2 & 1 & -2 \\ 1 & 2 & 3 \end{bmatrix}, \qquad B = \begin{bmatrix} 1 & 0 & 4 \\ 2 & 3 & -1 \\ 0 & 1 & 2 \end{bmatrix}$

add the corresponding entries:

$A + B = \begin{bmatrix} 1+1 & 3+0 & 4+4 \\ 2+2 & 1+3 & -2+(-1) \\ 1+0 & 2+1 & 3+2 \end{bmatrix} = \begin{bmatrix} 2 & 3 & 8 \\ 4 & 4 & -3 \\ 1 & 3 & 5 \end{bmatrix}$

Worked example: subtraction

Given

$A = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}, \qquad B = \begin{bmatrix} 3 & 4 \\ 6 & 5 \end{bmatrix}$

subtract the corresponding entries:

$A - B = \begin{bmatrix} 1-3 & 2-4 \\ 2-6 & 4-5 \end{bmatrix} = \begin{bmatrix} -2 & -2 \\ -4 & -1 \end{bmatrix}$

Multiplying a matrix by a scalar

A scalar is any constant. To multiply a matrix by a scalar, multiply every element by that constant.

For $A = \begin{bmatrix} 2 & 3 & 4 \end{bmatrix}$,

$3A = 3\begin{bmatrix} 2 & 3 & 4 \end{bmatrix} = \begin{bmatrix} 6 & 9 & 12 \end{bmatrix}$

Properties of matrix addition

Commutative. For any two matrices of the same order, $A + B = B + A$.

Associative. For three matrices $A$, $B$, $C$ of the same order, $(A + B) + C = A + (B + C)$.

Additive identity. The zero matrix is the additive identity: adding it leaves a matrix unchanged. For example, the additive identity of $\begin{bmatrix} 3 & 4 & 8 \end{bmatrix}$ is $\begin{bmatrix} 0 & 0 & 0 \end{bmatrix}$.

Additive inverse. If $A + B$ equals the zero matrix, then $B$ is the additive inverse of $A$ (and $A$ is the additive inverse of $B$). To find the additive inverse of a matrix, change the sign of every element. For example, the additive inverse of

$\begin{bmatrix} 3 & 2 \\ 0 & -1 \end{bmatrix} \quad \text{is} \quad \begin{bmatrix} -3 & -2 \\ 0 & 1 \end{bmatrix}$

Example: proving the associative property

Let

$P = \begin{bmatrix} 2 & 4 \\ 5 & 6 \end{bmatrix}, \qquad Q = \begin{bmatrix} 3 & 5 \\ -2 & 1 \end{bmatrix}, \qquad R = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}$

Left side. First $P + Q$, then add $R$:

$P + Q = \begin{bmatrix} 5 & 9 \\ 3 & 7 \end{bmatrix}, \qquad (P + Q) + R = \begin{bmatrix} 6 & 9 \\ 5 & 8 \end{bmatrix}$

Right side. First $Q + R$, then add $P$:

$Q + R = \begin{bmatrix} 4 & 5 \\ 0 & 2 \end{bmatrix}, \qquad P + (Q + R) = \begin{bmatrix} 6 & 9 \\ 5 & 8 \end{bmatrix}$

Both sides give the same matrix, so $(P + Q) + R = P + (Q + R)$, confirming the associative property.

Key takeaways

• Matrices can only be added or subtracted when they have the same order, by combining corresponding entries.
• Multiplying a matrix by a scalar multiplies every entry by that constant.
• Matrix addition is commutative and associative, the zero matrix is the additive identity, and the additive inverse is found by changing the sign of every entry.