Matrices

Chapter in a nutshell: A matrix is just a rectangular grid of numbers arranged in rows and columns — a neat way to handle many numbers at once (think of a table of marks). This topic is new, so go step by step: learn the order, the types, then the four operations (add, subtract, multiply by a number, multiply two matrices). The only tricky one is matrix × matrix — once you master the row × column rule, the rest is easy.

1. What is a matrix? (start here)

• A matrix is numbers arranged in a rectangle of rows (go across →) and columns (go down ↓), inside brackets, e.g. $A=\begin{bmatrix}2&5\\1&7\end{bmatrix}$.
• Each number is an element. The element in row $i$, column $j$ is $a_{ij}$. (In $A$, $a_{12}=5$.)
• Order $=$ (number of rows) $\times$ (number of columns) $=m\times n$ — rows first, always. $A$ above has order $2\times2$.
• Example: $\begin{bmatrix}3&0&-1\end{bmatrix}$ has order $1\times3$ (1 row, 3 columns).

2. Types of matrices (with examples)

• Row matrix — one row: $\begin{bmatrix}2&5\end{bmatrix}$ (order $1\times2$).
• Column matrix — one column: $\begin{bmatrix}2\\5\end{bmatrix}$ (order $2\times1$).
• Square matrix — rows $=$ columns: $\begin{bmatrix}1&2\\3&4\end{bmatrix}$.
• Null / zero matrix ($O$) — every element 0: $\begin{bmatrix}0&0\\0&0\end{bmatrix}$.
• Diagonal matrix — square, all non-diagonal elements 0.
• Identity / unit matrix ($I$) — diagonal matrix with 1s on the diagonal: $I=\begin{bmatrix}1&0\\0&1\end{bmatrix}$. (It behaves like the number 1.)
• Transpose ($A^T$) — rows and columns swapped.

3. Equal matrices

• Worked: if $\begin{bmatrix}x&3\\1&y\end{bmatrix}=\begin{bmatrix}2&3\\1&5\end{bmatrix}$, then $x=2$ and $y=5$.

4. The four operations — step by step

(a) Addition (only if same order) — add corresponding elements: $\begin{bmatrix}1&2\\3&4\end{bmatrix}+\begin{bmatrix}5&6\\7&8\end{bmatrix}=\begin{bmatrix}1+5&2+6\\3+7&4+8\end{bmatrix}=\begin{bmatrix}6&8\\10&12\end{bmatrix}$.

(b) Subtraction (only if same order) — subtract corresponding elements: $\begin{bmatrix}5&6\\7&8\end{bmatrix}-\begin{bmatrix}1&2\\3&4\end{bmatrix}=\begin{bmatrix}4&4\\4&4\end{bmatrix}$.

(c) Scalar multiplication — multiply every element by the number: $3\begin{bmatrix}1&-2\\0&4\end{bmatrix}=\begin{bmatrix}3&-6\\0&12\end{bmatrix}$.

(d) Matrix × Matrix (the important one):

1. Is it allowed? $A\times B$ exists only if columns of $A$ = rows of $B$.
2. Order of the answer: $(m\times n)\times(n\times p)=(m\times p)$ — the inner $n$'s match and "cancel".
3. How: each answer-element $=$ (a row of $A$) · (a column of $B$) $=$ multiply pair-by-pair and add.

Worked ($2\times2$ by $2\times2$): $A=\begin{bmatrix}1&2\\3&4\end{bmatrix},\ B=\begin{bmatrix}5&6\\7&8\end{bmatrix}$.

• Top-left $=(1)(5)+(2)(7)=5+14=19$
• Top-right $=(1)(6)+(2)(8)=6+16=22$
• Bottom-left $=(3)(5)+(4)(7)=15+28=43$
• Bottom-right $=(3)(6)+(4)(8)=18+32=50$
• So $AB=\begin{bmatrix}19&22\\43&50\end{bmatrix}$.

Worked ($2\times2$ by $2\times1$): $\begin{bmatrix}1&2\\3&4\end{bmatrix}\begin{bmatrix}5\\6\end{bmatrix}=\begin{bmatrix}1(5)+2(6)\\3(5)+4(6)\end{bmatrix}=\begin{bmatrix}17\\39\end{bmatrix}$ (order $2\times1$).

(e) Identity check: $A\times I=I\times A=A$ — e.g. $\begin{bmatrix}1&2\\3&4\end{bmatrix}\begin{bmatrix}1&0\\0&1\end{bmatrix}=\begin{bmatrix}1&2\\3&4\end{bmatrix}$.

5. Key properties (remember)

• Addition is commutative: $A+B=B+A$. Multiplication is NOT: usually $AB\ne BA$.
• $A+O=A$; $A\times I=A$.
• Multiplication is associative $(AB)C=A(BC)$ and distributive $A(B+C)=AB+AC$.
• $(A^T)^T=A$; $(A+B)^T=A^T+B^T$.

6. Common mistakes to avoid

• Writing order as columns×rows — it is rows × columns.
• Adding/subtracting matrices of different orders (not allowed).
• Doing matrix multiplication element-by-element — wrong; use row · column.
• Trying $A\times B$ when cols($A$) ≠ rows($B$).
• Assuming $AB=BA$.
• Forgetting the product's order is $(m\times p)$.

7. Likely exam questions (with crisp answers)

1. Order of $\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}$? → $2\times3$.
2. When can two matrices be added? → Same order.
3. Find $x,y$ from equal matrices? → equate corresponding elements.
4. Condition for $A\times B$ to exist? → cols($A$) = rows($B$).
5. Order of $(2\times3)(3\times4)$? → $2\times4$.
6. Is $AB=BA$ always? → No (not commutative).
7. What is $A\times I$? → $A$.
8. Define null matrix / identity matrix. → all-zero matrix / diagonal of 1s.
9. Find $2A-B$? → scalar-multiply, then subtract.
10. Compute a $2\times2$ product. → use the row · column method.

8. Prerequisite skills

• Arithmetic of signed numbers ($+,-,\times$).
• Distributive law $a(b+c)=ab+ac$.
• Reading an ordered array / table of numbers.