Matrix Addition and Subtraction

First, two matrices are identical if they have the same order (dimensions) and $b_{ij} = a_{ij}$. Intuitively, if the matrices have the same shape, and all of the elements in $\mathbf{A}$ are the same as all of the elemtns in $\mathbf{B}$ the matrices are equal.

If two matrices have the same order, it is possible to apply element-wise additional and subtraction operations. For example:

$\mathbf{A} = \begin{bmatrix} 10 & 2\\ 4 & 6 \\ 1 & 9 \end{bmatrix}$

and

$\mathbf{B} = \begin{bmatrix} 5 & 4\\ 2 & 1 \\ 8 & 3 \end{bmatrix}$

then

• $\mathbf{A} + \mathbf{B} = \begin{bmatrix} 10 + 5 & 2 + 4\\ 4 + 2 & 6 + 1 \\ 1 + 8 & 9 + 3 \end{bmatrix} = \begin{bmatrix} 15 & 6\\ 6 & 7 \\ 9 & 12\end{bmatrix}$

• $\mathbf{A} - \mathbf{B} = \begin{bmatrix} 10 - 5 & 2 - 4\\ 4 - 2 & 6 - 1 \\ 1 - 8 & 9 - 3 \end{bmatrix} = \begin{bmatrix} 5 & -2\\ 2 & 5 \\ -7 & 6\end{bmatrix}$

Also implicit in the above: $\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}$.

We can also identify a special matrix - the zero matrix.

$\mathbf{C} = \begin{bmatrix} 0 & 0\\ 0 & 0\\ 0 & 0 \end{bmatrix}$

Using the zero matrix, it is also possible to define:

$\mathbf{A} + \mathbf{C} = \mathbf{A}$, assuming that \textbf{A} and textbf{C} are the same order.

Supplemental Video

Multiplication by a scalar

It is possible to multiple a scalar with a matrix. For any scalar $\lambda_{1}$,

$\lambda_{1}\mathbf{A}\begin{bmatrix} \lambda_{1}a_{11} & \lambda_{1}a_{21}\\ \lambda_{1}a_{12} & \lambda_{1}a_{22} \\ \lambda_{1}a_{13} & \lambda_{1}a_{23} \end{bmatrix}$

and

$\lambda_{1}\mathbf{A} = \mathbf{A}\lambda_{1}$

For example:

$3\mathbf{A} = \begin{bmatrix} 3(10) & 3(2)\\ 3(4) & 3(6) \\ 3(1) & 3(9)\end{bmatrix} = \begin{bmatrix} 30 & 6\\ 12 & 18 \\ 3 & 27 \end{bmatrix}$

The final example illustrates the element-wise multiplication of a matrix with a scalar. This is not matrix multiplication.