3.3 Operations on Matrices

3.3 Operations on Matrices

Addition

Two matrices $A$ and $B$ of the same order can be added:

$$(A + B)_{ij} = a_{ij} + b_{ij}$$

Scalar Multiplication

$$(kA)_{ij} = k \cdot a_{ij}$$

Matrix Multiplication

If $A$ is $m \times n$ and $B$ is $n \times p$, then $AB$ is $m \times p$:

$$(AB)_{ij} = \sum_{k=1}^{n} a_{ik} \cdot b_{kj}$$

Note: In general, $AB \neq BA$. Matrix multiplication is not commutative.

Example

Let $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$.

$$AB = \begin{bmatrix} 1 \times 5 + 2 \times 7 & 1 \times 6 + 2 \times 8 \\ 3 \times 5 + 4 \times 7 & 3 \times 6 + 4 \times 8 \end{bmatrix} = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$$