Add and Subtract Matrices

It is easy to add and subtract matrices. If you know how to add and subtract integers, this lesson will be a piece of cake. First, study the figure below carefully to see how it is done for a 2 × 2 matrix.

Based on the example in the figure above, to add or subtract matrices, we add or subtract their corresponding elements. For example, we add 8 and 9 to get 17.

Given two m × n matrices A = [ a_ij ] and B = [ b_ij ], with a and b located in row i and column j.

The sum is A + B = [ a_ij ] + [ b_ij ]

The difference is A - B = [ a_ij ] - [ b_ij ]

Using two 2 × 2 matrices,

• the elements in the first row and first column are a₁₁ and b₁₁

• the elements in the first row and second column are a₁₂ and b₁₂

• the elements in the second row and first column are a₂₁ and b₂₁

• the elements in the second row and second column are a₂₂ and b₂₂

Notice how we get the sum of the corresponding elements in the following addition of matrices:

$$ A + B = \begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12}\\ b_{21} & b_{22}\\ \end{bmatrix} = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12}\\ a_{21} + b_{21} & a_{22} + b_{22}\\ \end{bmatrix} $$

Notice also how we get the difference of the corresponding elements in the following subtraction of matrices:

$$ A - B = \begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix} - \begin{bmatrix} b_{11} & b_{12}\\ b_{21} & b_{22}\\ \end{bmatrix} = \begin{bmatrix} a_{11} - b_{11} & a_{12} - b_{12}\\ a_{21} - b_{21} & a_{22} - b_{22}\\ \end{bmatrix} $$

In order words, you can add or subtract a 2x3 with a 2x3 or a 3x3 with a 3x3. However, you cannot add a 3x2 with a 2x3 or a 2x2 with a 3x3.

More examples showing how to add and subtract matrices

Rule to follow in order to add and subtract matrices

You can add elements in the same position or elements in the same row and same column. We show these elements with the same color to make this crystal clear.

Matrix subtraction