Chapter 3 at a Glance

The objects

A matrix is an ordered rectangular array of numbers or functions. A matrix with $m$ rows and $n$ columns has order $m \times n$ and $mn$ entries; the entry in row $i$, column $j$ is written $a_{ij}$.

Equality: $A = B$ requires the same order and $a_{ij} = b_{ij}$ for every position.

The operations

Addition (same order only): entrywise; commutative and associative, with $A + O = A$ and $A + (-A) = O$.

Scalar multiple: $kA = [k a_{ij}]$, with $k(A+B) = kA + kB$ and $(k + l)A = kA + lA$.

Multiplication ($A$ is $m \times n$, $B$ is $n \times p$, product $m \times p$): row-by-column, $c_{ik} = \sum_{j=1}^{n} a_{ij} b_{jk}$.

Multiplication is associative and distributes over addition, and $AI = IA = A$ — but two familiar rules break:

1. $AB \neq BA$ in general — order matters.
2. $AB = O$ does not force $A = O$ or $B = O$ — nonzero zero divisors exist, so you cannot cancel a matrix factor.

Transpose, symmetric, skew-symmetric

$A'$ (or $A^T$) swaps rows with columns: if $A$ is $m \times n$ then $A'$ is $n \times m$.

$$(A')' = A, \qquad (kA)' = kA', \qquad (A + B)' = A' + B', \qquad (AB)' = B'A'$$

The reversal in the last law is the one students forget — it mirrors $(AB)^{-1} = B^{-1}A^{-1}$.

Symmetric: $A' = A$. Skew-symmetric: $A' = -A$ (forcing every diagonal entry to be $0$). Every square matrix splits uniquely as

$$A = \underbrace{\tfrac{1}{2}(A + A')}_{\text{symmetric}} + \underbrace{\tfrac{1}{2}(A - A')}_{\text{skew-symmetric}}$$

Invertible matrices

If $A$ and $B$ are square matrices of the same order with $AB = BA = I$, then $B = A^{-1}$ (and $A = B^{-1}$). The inverse, when it exists, is unique — proof in one line: $B = BI = B(AC) = (BA)C = IC = C$.

JEE favourite: a polynomial identity like $A^2 - 2A - 3I = O$ rearranges to $A \cdot \frac{1}{3}(A - 2I) = I$, handing over $A^{-1}$ with no formula.

Mistake checklist

Before the exam, make sure none of these can catch you:

1. Writing $AB$ for $BA$ — always check which side a multiplication acts on; expanding $(A+B)^2$ as $A^2 + 2AB + B^2$ silently assumes $AB = BA$.
2. Cancelling matrices — from $AB = AC$ you may not conclude $B = C$ unless $A$ is invertible.
3. Forgetting the reversal laws — $(AB)' = B'A'$ and $(AB)^{-1} = B^{-1}A^{-1}$, both reversed.
4. Adding matrices of different orders — undefined, even if it "looks close".
5. Counting orders — a matrix with $12$ entries has $6$ possible orders (ordered pairs!), and a $3 \times 3$ matrix with entries from $\{0, 1\}$ can be built in $2^9 = 512$ ways.
6. Diagonal of a skew-symmetric matrix — always all zeros; its entries sum to $0$.
7. Huge exponents — never multiply out $A^{2023}$; hunt for $A^2 = I$, $A^2 = A$, $A^2 = kA$ or $A^k = O$ first.

The 15 questions below are a fast pass over the whole chapter — everything at recall level. If any of them feels shaky, the section named in its explanation is where to go back.