Algebra of Matrices — Equality, Scalar Multiplication and Addition

4.5 Algebra of Matrices — Equality, Scalar Multiplication, Addition

Four operations make up the algebra of matrices: (1) equality, (2) multiplication by a scalar, (3) addition, (4) multiplication of two matrices (the last one is big enough to need its own section, §4.5's continuation below).

(1) Equality of matrices. $A=B$ iff (i) $A$ and $B$ have the same order, AND (ii) every corresponding pair of elements matches: $a_{ij}=b_{ij}$ for all $i,j$.

(2) Multiplication of a matrix by a scalar. For $A=[a_{ij}]_{m\times n}$ and a scalar $k$, $kA=[k\,a_{ij}]_{m\times n}$ — every entry is scaled by $k$; the order is unchanged.

(3) Addition of two matrices. For $A=[a_{ij}]_{m\times n}$ and $B=[b_{ij}]_{m\times n}$ of the same order, $A+B=[a_{ij}+b_{ij}]_{m\times n}$ — add corresponding entries; $A+B$ has the same order as $A,B$. (Addition is undefined for matrices of different orders.) Subtraction: $A-B=A+(-B)$, where $-B$ negates every entry of $B$.

Algebraic laws (for matrices $A,B,C$ conformable for addition, scalars $\alpha,\beta$)

1. $A+B=B+A$ — addition is commutative.
2. $(A+B)+C=A+(B+C)$ — addition is associative.
3. $A+O=O+A=A$ — the zero matrix $O$ is the additive identity.
4. $A+(-A)=(-A)+A=O$ — $(-A)$ is the additive inverse of $A$.
5. $\alpha(A\pm B)=\alpha A\pm\alpha B$.
6. $(\alpha\pm\beta)A=\alpha A\pm\beta A$.
7. $\alpha(\beta A)=(\alpha\beta)A$.
8. $OA=O$.

Worked Examples

Example (equality). If $\begin{bmatrix}2a-b&-7\\2&4\end{bmatrix}=\begin{bmatrix}a+1&-7\\4+b&3\end{bmatrix}$ — wait, more simply: given $2a-b=1$ and $a+3b=2$ from matching two entries, solving simultaneously gives $a=\dfrac57,\ b=\dfrac37$.

Example (scalar multiplication). If $A=\begin{bmatrix}1&2&4\\5&4&7\end{bmatrix}$ and $k=\dfrac32$, then $kA=\dfrac32A=\begin{bmatrix}\tfrac32&3&6\\\tfrac{15}2&6&\tfrac{21}2\end{bmatrix}$ — every entry multiplied by $\tfrac32$.

Example 1. If $A=\begin{bmatrix}5&3&-1\\0&4&-2\end{bmatrix}$ and $B=\begin{bmatrix}-2&-7&3\\1&-2&2\end{bmatrix}$, find $2A-3B$.

Step 1: $2A=\begin{bmatrix}10&6&-2\\0&8&-4\end{bmatrix}$, $3B=\begin{bmatrix}-6&-21&9\\3&-6&6\end{bmatrix}$. …