Matrices and Determinants

A matrix is a rectangular array of numbers arranged in rows and columns. An m×n matrix has m rows and n columns.

Square matrix: m = n.
Row matrix: 1×n.
Column matrix: m×1.
Diagonal matrix: square, non-zero entries only on the diagonal.
Identity matrix (I): diagonal entries 1, others 0. I_n is the n×n identity.
Zero matrix: all entries zero.

OPERATIONS

Addition / subtraction: element-wise. Matrices must be the same size.

Scalar multiplication: k · A means multiply every entry by k.

Matrix multiplication: (AB)_ij = Σ_k A_ik B_kj. A must be m×n, B must be n×p, result is m×p.

• Not commutative: AB ≠ BA in general.
• Associative: (AB)C = A(BC).
• Distributive: A(B + C) = AB + AC.

Transpose A^T (or A'): flip rows and columns. (A^T)_ij = A_ji.

Properties:

• (A^T)^T = A
• (A + B)^T = A^T + B^T
• (AB)^T = B^T A^T (note the order reversal!)
• (kA)^T = k A^T

SPECIAL SQUARE MATRICES

Symmetric: A^T = A. (entries symmetric across diagonal)
Skew-symmetric: A^T = −A. (diagonal entries are 0; off-diagonal mirror with opposite sign)

Property: every square matrix can be uniquely written as the sum of a symmetric and a skew-symmetric matrix:
A = (A + A^T)/2 + (A − A^T)/2.

Orthogonal: A^T A = I. Columns (and rows) are orthonormal.
Hermitian (complex): A* = A (conjugate transpose).
Idempotent: A² = A.
Nilpotent: A^k = 0 for some k.
Involutory: A² = I.

INVERSE

A square matrix A has an inverse A⁻¹ if A · A⁻¹ = A⁻¹ · A = I.

Existence: A is invertible iff det(A) ≠ 0 (non-singular).

Formula (for 2×2):
A = [[a, b], [c, d]], det(A) = ad − bc.
A⁻¹ = (1/det A) · [[d, −b], [−c, a]].

General formula:
A⁻¹ = (1/det A) · adj(A)

where adj(A) = transpose of the cofactor matrix.

Properties:

• (A⁻¹)⁻¹ = A
• (AB)⁻¹ = B⁻¹ A⁻¹ (order reversal again!)
• (A^T)⁻¹ = (A⁻¹)^T
• det(A⁻¹) = 1 / det(A)
• (kA)⁻¹ = (1/k) A⁻¹

SOLVING LINEAR SYSTEMS Ax = b:

If A is invertible: x = A⁻¹ b.

Or use Cramer's rule: x_i = det(A_i) / det(A), where A_i is A with column i replaced by b.

For singular A (det = 0):

• Infinitely many solutions if rank(A) = rank([A|b]).
• No solution if rank(A) < rank([A|b]) (inconsistent).

Worked example. A = [[2, 1], [3, 4]]. Find A⁻¹.

det = 8 − 3 = 5. A⁻¹ = (1/5) · [[4, −1], [−3, 2]] = [[0.8, −0.2], [−0.6, 0.4]].

Check: A · A⁻¹ = [[2(0.8) + 1(−0.6), 2(−0.2) + 1(0.4)], [3(0.8) + 4(−0.6), 3(−0.2) + 4(0.4)]] = [[1, 0], [0, 1]] = I. ✓

Determinant of a square matrix A is a scalar denoted det(A) or |A|.

For 2×2: | a b ; c d | = ad − bc.

For 3×3 (cofactor expansion along first row):
| a b c ; d e f ; g h i | = a(ei − fh) − b(di − fg) + c(dh − eg).

Properties (each one is a JEE shortcut):

1. Swap of two rows (or columns) → determinant changes sign.

2. If two rows are identical (or proportional) → determinant = 0.

3. Multiplying a row by k → determinant multiplies by k. Therefore det(kA) = k^n · det(A) for n×n matrix.

4. Adding a scalar multiple of one row to another → determinant unchanged. (This is the basis of row reduction.)

5. det(A^T) = det(A). All row properties apply to columns too.

6. det(AB) = det(A) · det(B). Crucial: det of product = product of dets.

7. det(A⁻¹) = 1 / det(A). Therefore A is invertible iff det(A) ≠ 0.

8. Triangular matrix: determinant = product of diagonal entries.

Cramer's rule for solving Ax = b (when det(A) ≠ 0):

x_i = det(A_i) / det(A)

where A_i is A with the i-th column replaced by b.

Worked example. Find det of:
| 2 3 1 ; 4 6 2 ; 5 7 9 |

Notice row 2 = 2 × row 1 — so determinant = 0 (Property 2). Saved a 3-minute cofactor expansion.