Matrices — review, operations, types: Worked examples

The scenario matrix

Think of every matrix exercise as landing in one of these boxes. If you can do one example from each box, nothing on an exam is new.

Cell	Scenario	What makes it tricky
A	Compatible product $A B$ , both nonzero	getting the row·column pairing right
B	Non-commutativity: compute $A B$ and $B A$	showing they differ
C	Degenerate: multiplying by the zero matrix / identity	the "do nothing" and "collapse" cases
D	Zero divisors: $A B = 0$ with $A \neq = 0, B \neq = 0$	the surprising box that breaks number-intuition
E	Transpose reversal $(A B)^{⊤} = B^{⊤} A^{⊤}$	order flips
F	Symmetric + skew split	the $\frac{1}{2}$ trick, zero diagonal
G	Shape mismatch: when a product simply does not exist	reading dimensions
H	Limiting / repeated: powers $A^{2}, A^{3}$ of a matrix	composition with itself
I	Word problem: matrices modelling something real	translating words to a grid
J	Exam twist: orthogonal matrix, verify $A^{⊤} A = I$	length-preserving check

Every symbol below is built from scratch. A matrix is just a rectangular grid of numbers; an entry $a_{ij}$ sits in row $i$ , column $j$ (count rows top-to-bottom, columns left-to-right). The dot product of a row and a column means: multiply matching numbers and add them up — that single idea powers every product on this page. See it as sliding a horizontal strip onto a vertical strip:

Figure — Matrices — review, operations, types

Cell A — a compatible product

Worked example Compute

$A B$ $A = [20 - 1 3], B = [12 4 - 2]$

Forecast: before reading on, guess the top-left entry. It is row 1 of $A$ = $(2, - 1)$ dotted with column 1 of $B$ = $(1, 2)$ . Hold your guess.

Step 1 — check the shapes. Why this step? A product only exists if the inner dimensions match: $A$ is $2 \times 2$ , $B$ is $2 \times 2$ , inner numbers are both $2$ . ✓ Result is $2 \times 2$ .

Step 2 — entry $(1, 1)$ : row 1 · col 1. $2 \cdot 1 + (- 1) \cdot 2 = 2 - 2 = 0$ Why this step? Each output entry is one row met with one column; nothing else is combined.

Step 3 — remaining entries. $(1, 2) : 2 \cdot 4 + (- 1) (- 2) = 8 + 2 = 10$ $(2, 1) : 0 \cdot 1 + 3 \cdot 2 = 6$ $(2, 2) : 0 \cdot 4 + 3 \cdot (- 2) = - 6$ $A B = [06 10 - 6]$

Verify: the $(1, 1)$ entry landed on $0$ — a coincidence here, not a rule. Sanity check: row 2 of $A$ starts with $0$ , so both entries of row 2 of $A B$ come only from the $3$ : $3 \cdot (2, - 2) = (6, - 6)$ . ✓ Matches.

Cell B — order matters ( $A B \neq = B A$ )

Cell C — the degenerate multipliers: $I$ and $0$

Cell D — zero divisors (the surprising box)

Cell E — transpose reversal $(A B)^{⊤} = B^{⊤} A^{⊤}$

Cell F — symmetric + skew split

Cell G — the mismatch (product does not exist)

Cell H — powers (composition with itself)

Cell I — a real-world word problem

Cell J — exam twist: verify orthogonality

Worked example Is this rotation matrix orthogonal?

$R = [cos θ sin θ - sin θ cos θ]$ A matrix is orthogonal when $R^{⊤} R = I$ ; such maps preserve lengths — pure rotations/reflections (central in Eigenvalues and Eigenvectors).

Forecast: guess the product $R^{⊤} R$ before expanding — should it be $I$ ?

Step 1 — build $R^{⊤} R$ . $R^{⊤} = [cos θ - sin θ sin θ cos θ]$ . $(1, 1) : cos^{2} θ + sin^{2} θ, (1, 2) : - cos θ sin θ + sin θ cos θ, \dots$ Why this step? Each entry is a dot product of two columns of $R$ ; orthogonality means columns are unit-length and perpendicular.

Step 2 — simplify using $cos^{2} θ + sin^{2} θ = 1$ . $R^{⊤} R = [1001] = I$

Verify (numeric, $θ = 3 0^{\circ}$ ): $cos 3 0^{\circ} = \frac{3}{2}, sin 3 0^{\circ} = \frac{1}{2}$ . Then $(1, 1) = \frac{3}{4} + \frac{1}{4} = 1$ ✓ and the off-diagonal $- \frac{3}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot \frac{3}{2} = 0$ ✓. Rotation preserves length as expected.

Recall Which cell was which?

Every product exists only when ::: the inner dimensions match (columns of first = rows of second). The box that breaks number-intuition ::: zero divisors, $P Q = 0$ with $P, Q \neq = 0$ (Cell D). Transpose of a product ::: reverses order: $(A B)^{⊤} = B^{⊤} A^{⊤}$ (Cell E). Skew part always has ::: a zero diagonal (Cell F). $A^{n}$ for the shear $[1011]$ ::: $[10 n 1]$ (Cell H). Orthogonal test ::: $R^{⊤} R = I$ (Cell J).

Connections

Matrices — review, operations, types — the parent note these examples deepen.
Dot Product and Vectors — every product entry is one of these (Cells A, I).
Linear Transformations — powers and shears as repeated maps (Cell H).
Determinant and Inverse of a Matrix — why zero divisors block cancellation (Cell D).
Eigenvalues and Eigenvectors — orthogonal matrices reappear (Cell J).
Systems of Linear Equations — the compression motivating all of this.