Adding Matrices & Subtracting Matrices

Adding Matrices & Subtracting Matrices

Matrices are rectangular arrays of numbers, which can be used to represent linear transformations, data sets, and many other mathematical objects. In this article, we will discuss how to add and subtract matrices and provide some examples to illustrate the concepts.

Adding Matrices

To add two matrices, we simply add the corresponding entries of the matrices. For example, given two matrices A and B, we can add them by performing the following operation:

C = 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}

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Note that in order to add two matrices, they must have the same size, i.e., they must have the same number of rows and columns. If the matrices do not have the same size, it is not possible to add them.

For example, consider the following two matrices:

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

B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}

We can add these two matrices as follows:

C = A + B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}

Subtracting Matrices

Subtracting matrices is similar to adding matrices, but we subtract the corresponding entries instead of adding them. For example, given two matrices A and B, we can subtract them by performing the following operation:

C = 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}

As with addition, in order to subtract two matrices, they must have the same size.

For example, consider the following two matrices:

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

B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}

We can subtract these two matrices as follows:

C = A - B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} - \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 1-5 & 2-6 \\ 3-7 & 4-8 \end{bmatrix} = \begin{bmatrix} -4 & -4 \\ -4 & -4 \end{bmatrix}

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