Matrices & Determinants Notes

Last reviewed 16 Sept 2026 · 7 min read

Matrices

A matrix is a rectangular array of numbers arranged in m rows and n columns (order $m \times n$ ).

Type	Definition
Row / column matrix	Single row / single column
Square matrix	$m = n$
Diagonal matrix	Square; all off-diagonal elements zero
Scalar matrix	Diagonal with all diagonal elements equal ( $k I$ )
Identity (unit) matrix $I$	Diagonal elements 1, others 0
Null (zero) matrix	All elements zero
Upper / lower triangular	All elements below / above the diagonal are zero
Symmetric	$A^{T} = A$ ( $a_{ij} = a_{j i}$ )
Skew-symmetric	$A^{T} = - A$ — diagonal elements are zero
Orthogonal	$A A^{T} = A^{T} A = I$ → $A^{- 1} = A^{T}$ , $∣ A ∣ = \pm 1$
Idempotent	$A^{2} = A$
Nilpotent	$A^{k} = O$ for some positive integer $k$
Involutory	$A^{2} = I$ ( $A^{- 1} = A$ )
Singular / non-singular	$∣ A ∣ = 0$ / $∣ A ∣ \neq = 0$
Hermitian	$A^{θ} = A$ (conjugate transpose equals itself)
Skew-Hermitian	$A^{θ} = - A$
Unitary	$A A^{θ} = I$

Every square matrix can be written uniquely as the sum of a symmetric and a skew-symmetric matrix:

A = \frac{1}{2} (A + A^{T}) + \frac{1}{2} (A - A^{T})

Addition — same order, element-wise; commutative and associative.
Scalar multiplication — multiply every element.
Multiplication $A B$ — defined when columns of A = rows of B; $(A B)_{ij} = \sum_{k} a_{ik} b_{k j}$ ; not commutative in general ( $A B \neq = B A$ ); associative; distributive.
$A B = O$ does not imply $A = O$ or $B = O$ .

$(A^{T})^{T} = A$ ; $(A + B)^{T} = A^{T} + B^{T}$ ; $(k A)^{T} = k A^{T}$ ; $(A B)^{T} = B^{T} A^{T}$ .

For a $2 \times 2$ matrix: $a c b d = a d - b c$ .

For a $3 \times 3$ matrix, expand along any row or column using cofactors:

∣ A ∣ = a_{11} C_{11} + a_{12} C_{12} + a_{13} C_{13}, C_{ij} = (- 1)^{i + j} M_{ij}

( $M_{ij}$ = minor — determinant after deleting row $i$ and column $j$ .)

$∣ A^{T} ∣ = ∣ A ∣$ .
Interchanging two rows (or columns) changes the sign.
Two identical (or proportional) rows → determinant zero.
Multiplying a row by $k$ multiplies the determinant by $k$ ; hence $∣ k A ∣ = k^{n} ∣ A ∣$ for an $n \times n$ matrix.
Adding a multiple of one row to another does not change the determinant.
$∣ A B ∣ = ∣ A ∣∣ B ∣$ .
Determinant of a triangular or diagonal matrix = product of diagonal elements.
Sum of products of elements of a row with cofactors of another row = 0.

FormulaInverse and related identities (A of order n)

A^{- 1} = \frac{adj A}{∣ A ∣}, ∣ A ∣ \neq = 0

For $2 \times 2$ : $[a c b d]^{- 1} = \frac{1}{a d - b c} [d - c - b a]$

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