How to Subtract Matrices

To subtract two matrices, we simply subtract the numbers that are in like entry positions. Matrices must have the same dimension to be subtractable.

Here is what matrix subtraction for 2×2 matrices looks like:

$$\begin{align}& \begin{bmatrix} a1 & b1\\ c1 & d1 \end{bmatrix} - \begin{bmatrix} a2 & b2\\ c2 & d2 \end{bmatrix} = \begin{bmatrix} a1 - a2 & b1 - b2\\ c1 - c2 & d1 - d2 \end{bmatrix} \end{align}$$

Rules of Matrix Subtraction

Matrix subtraction and matrix addition have the same rules. The operation for subtracting 2×3 matrices follows the same process as subtracting 2×2 matrices. The operation for subtracting 3×3 matrices follows the same process as well. The operation for subtracting 11×22 matrices also follows the same process.

In fact, we can subtract matrices of any size as long as the two matrices being subtracted have the same dimension (i.e. must both be 4×4 or must both be 2×9, etc). Make sure keep track of any negative numbers during matrix subtraction, they can get easily lost or sign flipped when working with a large matrix!

Matrix Subtraction Example Problem

Let's work through an example problem together to reinforce our skills with subtracting matrices.

Find the final matrix by subtracting the second matrix from the first matrix:

$$\begin{align}& \text{M1 } = \begin{bmatrix} 6 & 6\\ 10 & 6 \end{bmatrix} \hspace{3ex} \text{ M2 } = \begin{bmatrix} 5 & 1\\ 2 & 4 \end{bmatrix} \end{align}$$

Solution:

1. The matrices are both 2×2, so they meet the requirement of having the same dimension. Let’s subtract the second matrix from the first by subtracting the numbers in like entry positions.
2. a1 - a2 = 6 - 5 = 1
3. b1 - b2 = 6 - 1 = 5
4. c1 - c2 = 10 - 2 = 8
5. d1 - d2 = 6 - 4 = 2
6. Now let’s plug the numbers into our final matrix.

$$\begin{align}& \text{M1 } - \text{ M2 } = \begin{bmatrix} 1 & 5\\ 8 & 2 \end{bmatrix} \end{align}$$