Operations on matrices

Three operations turn the set of matrices into an arithmetic system: addition, scalar multiplication, and matrix multiplication. Addition and scalar multiplication are easy: combine entry by entry. Matrix multiplication is more involved , and its rules surprise students at first. This subtopic walks through each operation carefully and lists the algebraic laws they obey.

Addition

If $A = (a_{ij})$ and $B = (b_{ij})$ have the same order $m \times n$, their sum is defined entry-wise:

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

So $A + B$ also has order $m \times n$. Matrix addition is commutative and associative, has identity $O$ (the zero matrix), and every $A$ has inverse $-A$.

If $A$ and $B$ have different orders, $A + B$ is undefined.

Scalar multiplication

For a scalar $\lambda \in \mathbb{R}$ and matrix $A = (a_{ij})$, the product $\lambda A$ is defined by multiplying every entry by $\lambda$:

$(\lambda A)_{ij} = \lambda \cdot a_{ij}.$

So $\lambda A$ has the same order as $A$.

Properties: $\lambda(A + B) = \lambda A + \lambda B$, $(\lambda + \mu) A = \lambda A + \mu A$, $\lambda(\mu A) = (\lambda \mu) A$, $1 \cdot A = A$.

Matrix multiplication

For $A$ of order $m \times p$ and $B$ of order $p \times n$, the product $AB$ is the $m \times n$ matrix with

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

The number of columns of $A$ must equal the number of rows of $B$. Otherwise $AB$ is undefined.

In words: to compute the $(i, j)$ entry of $AB$, take the $i$-th row of $A$, the $j$-th column of $B$, multiply corresponding entries and add.

Worked computation

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

$(AB)_{11} = 1 \cdot 5 + 2 \cdot 7 = 19$. $(AB)_{12} = 1 \cdot 6 + 2 \cdot 8 = 22$. $(AB)_{21} = 3 \cdot 5 + 4 \cdot 7 = 43$. $(AB)_{22} = 3 \cdot 6 + 4 \cdot 8 = 50$.

So $AB = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}$. Compute $BA$ the same way: $BA = \begin{pmatrix} 23 & 34 \\ 31 & 46 \end{pmatrix}$. Note $AB \neq BA$.

Algebraic laws

Matrix addition and multiplication satisfy:

• Associativity of addition: $(A + B) + C = A + (B + C)$.
• Commutativity of addition: $A + B = B + A$.
• Zero matrix: $A + O = A$.
• Associativity of multiplication: $A(BC) = (AB)C$ whenever all products are defined.
• Distributivity: $A(B + C) = AB + AC$, $(A + B)C = AC + BC$.
• Identity matrix: $AI = IA = A$ for square $A$ of the right order.

Non-commutativity: in general $AB \neq BA$.

Zero divisors exist: $AB$ can be $O$ even if $A, B \neq O$. Example: $A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$, $B = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$, then $AB = O$. This makes cancellation laws fail.

Powers of a square matrix

For a square $A$, $A^2 = AA$, $A^3 = A \cdot A^2$, and so on. $A^0 = I$ by convention.

A matrix $A$ is called

• idempotent if $A^2 = A$,
• involutory if $A^2 = I$,
• nilpotent if $A^k = O$ for some positive integer $k$.

Worked examples

Example 1. Compute $A + B$ for $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, B = \begin{pmatrix} -1 & 1 \\ 0 & 2 \end{pmatrix}$.

$\begin{pmatrix} 0 & 3 \\ 3 & 6 \end{pmatrix}$.

Example 2. Compute $3A - 2B$ for the same $A, B$.

$3A = \begin{pmatrix} 3 & 6 \\ 9 & 12 \end{pmatrix}, 2B = \begin{pmatrix} -2 & 2 \\ 0 & 4 \end{pmatrix}$. Difference: $\begin{pmatrix} 5 & 4 \\ 9 & 8 \end{pmatrix}$.

Example 3. Multiply $A = \begin{pmatrix} 1 & 2 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix}$.

$AB$ is $1 \times 1$: $1 \cdot 4 + 2 \cdot 5 + 3 \cdot 6 = 32$. So $AB = (32)$. And $BA$ is $3 \times 3$: $BA = \begin{pmatrix} 4 & 8 & 12 \\ 5 & 10 & 15 \\ 6 & 12 & 18 \end{pmatrix}$.

Example 4. Show $A = \begin{pmatrix} 2 & -3 \\ 1 & -2 \end{pmatrix}$ is involutory.

$A^2 = \begin{pmatrix} 2(2) + (-3)(1) & 2(-3) + (-3)(-2) \\ 1(2) + (-2)(1) & 1(-3) + (-2)(-2) \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I$.

Example 5. Show $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ is nilpotent.

$A^2 = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} = O$. So $A$ is nilpotent of index $2$.

Example 6. Let $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$. Show $A^n = \begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix}$ by induction.

Base: $A^1 = A$ has $(1, 2)$-entry equal to $1$. Step: $A^{n+1} = A \cdot A^n = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & n + 1 \\ 0 & 1 \end{pmatrix}$. $\blacksquare$

Try it yourself

1. Compute $A + B$ for $A = \begin{pmatrix} 1 \\ 2 \end{pmatrix}, B = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$.
2. Compute $2A$ for $A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$.
3. Multiply $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\begin{pmatrix} 5 & 0 \\ 0 & 6 \end{pmatrix}$.
4. Multiply $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\begin{pmatrix} 1 \\ -1 \end{pmatrix}$.
5. Find $A^2$ for $A = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$.
6. Find $A^2$ for $A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$.
7. Verify: $A(B + C) = AB + AC$ for matrices $A = (1, 1), B = (1), C = (2)$. (Choose orders.)
8. Find a $2 \times 2$ matrix $A \ne O$ with $A^2 = O$.
9. Find a $2 \times 2$ matrix $A \ne I$ with $A^2 = I$.
10. Find $A^{10}$ for $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$.
11. Show that the sum of two symmetric matrices is symmetric.
12. Find $AB$ and $BA$ for $A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$.
13. Show $A^2 = 3A$ for $A = \begin{pmatrix} 3 & 0 \\ 0 & 0 \end{pmatrix}$.
14. Multiply $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\begin{pmatrix} \cos\phi & -\sin\phi \\ \sin\phi & \cos\phi \end{pmatrix}$ and identify the result.

Pitfalls / Tricks

• Matrix multiplication is not commutative: always check the order before declaring $AB = BA$.
• Always confirm dimensions before multiplying. $A_{2 \times 3}$ times $B_{2 \times 3}$ is undefined.
• $AB = O$ does not imply $A = O$ or $B = O$. There exist zero divisors.
• Powers of a matrix can be computed efficiently using induction (as in Example 6) or eigenvalues.
• The rotation matrix in problem 14 satisfies $R(\theta) R(\phi) = R(\theta + \phi)$ , a useful structural identity.

Next we look at the transpose and the symmetric/skew-symmetric decomposition.