Matrices, Determinants & Inverse of a Matrix
(KEAM Mathematics – Linear Algebra)
TL;DR
- A matrix is a rectangular array of numbers.
- The determinant of a square matrix is a single number that tells whether the matrix is invertible.
- If
det(A) ≠ 0, the inverseA⁻¹exists and can be found using cofactors or row‑reduction.
1. Simple Explanation
| Concept | What it means | Why it matters |
|---|---|---|
| Matrix | A list of numbers arranged in rows and columns, e.g. A = [[a₁₁, a₁₂], [a₂₁, a₂₂]]. |
Encodes linear transformations and systems of equations. |
| Determinant | A scalar value derived from a square matrix. For 2×2: det(A) = a₁₁a₂₂ – a₁₂a₂₁. |
If det(A) = 0, the matrix is singular (no inverse). |
| Inverse | A matrix A⁻¹ such that A·A⁻¹ = I (identity). |
Solves linear systems: Ax = b → x = A⁻¹b. |
2. Key Points
2.1 Matrix Notation & Types
| Symbol | Meaning | Example |
|---|---|---|
A |
General matrix | A_{m×n}: m rows, n columns |
I_n |
Identity matrix of size n |
I₃ = [[1,0,0],[0,1,0],[0,0,1]] |
O |
Zero matrix | O_{2×3} = [[0,0,0],[0,0,0]] |
D |
Diagonal matrix | D = diag(d₁,…,d_n) |
T |
Transpose | Aᵀ swaps rows & columns |
2.2 Matrix Operations
| Operation | Rule | Example |
|---|---|---|
| Addition/Subtraction | Add/subtract corresponding entries | [[1,2],[3,4]] + [[5,6],[7,8]] = [[6,8],[10,12]] |
| Scalar Multiplication | Multiply each entry by scalar | 3·[[1,2],[3,4]] = [[3,6],[9,12]] |
| Matrix Multiplication | C = A·B where C_{ij} = Σ_k A_{ik}B_{kj} |
[[1,2],[3,4]]·[[5,6],[7,8]] = [[19,22],[43,50]] |
| Transpose | Aᵀ_{ij} = A_{ji} |
[[1,2],[3,4]]ᵀ = [[1,3],[2,4]] |
2.3 Determinant of a Square Matrix
| Size | Formula | Shortcut |
|---|---|---|
| 2×2 | det(A) = a₁₁a₂₂ – a₁₂a₂₁ |
Direct |
| 3×3 | Sarrus Rule: sum of products along diagonals minus sum of anti‑diagonals | ` |
| n×n | Laplace Expansion (cofactor expansion) | det(A) = Σ_j a_{ij}C_{ij} for any fixed row/column i/j |
| Properties | ||
– det(AB) = det(A)·det(B) |
||
| – Swapping rows/columns flips sign | det(A') = –det(A) |
|
| – Adding multiple of one row to another does not change det | det(A') = det(A) |
|
| – If a row/column is all zeros, det = 0 |
2.4 Inverse of a Matrix
| Condition | Method | Formula |
|---|---|---|
det(A) ≠ 0 |
Cofactor‑Adjugate | A⁻¹ = (1/det(A)) · adj(A) |
| Row‑Reduction | Augment A with I and row‑reduce to `[I |
|
| Gauss‑Jordan | Same as above, but reduce to reduced row‑echelon form (RREF) |
Adjugate: transpose of cofactor matrix.
Cofactor C_{ij} = (–1)^{i+j}·M_{ij} where M_{ij} is minor (determinant of submatrix after removing row i and column j).