Matrices & Determinants
WBJEE – Mathematics
1. Simple Explanation
| Concept | What it is | Why it matters |
|---|---|---|
| Matrix | A rectangular array of numbers arranged in rows and columns. | Basic tool for representing linear systems, transformations, and many algebraic structures. |
| Determinant | A scalar value associated with a square matrix that encodes many of its properties (invertibility, volume scaling, etc.). | Used to decide if a matrix is invertible, solve linear equations (Cramer’s rule), and compute eigenvalues. |
| Rank | The maximum number of linearly independent rows (or columns) in a matrix. | Indicates the dimension of the column/row space; tells whether a linear system has a unique solution. |
| Inverse | For a square matrix (A), a matrix (A^{-1}) such that (AA^{-1}=I). | Needed to solve (AX=B) as (X=A^{-1}B). |
2. Key Points
| Topic | Key Points |
|---|---|
| Types of Matrices | • Square – same number of rows & columns. • Rectangular – rows ≠ columns. • Zero – all entries 0. • Identity (I_n) – 1’s on diagonal, 0’s elsewhere. • Diagonal – non‑zero only on diagonal. • Upper/Lower Triangular – all entries below/above main diagonal are 0. • Symmetric – (A = A^T). |
| Matrix Operations | • Addition/Subtraction – element‑wise; only for same size. • Scalar Multiplication – multiply every element by a scalar. • Matrix Multiplication – (C_{ij}=\sum_k A_{ik}B_{kj}); sizes must conform. • Transpose – (A^T_{ij}=A_{ji}). |
| Determinants | • Denoted (\det A) or ( |
| Cofactor & Adjugate | • Minor (M_{ij}): determinant of matrix after deleting row (i), column (j). • Cofactor (C_{ij}=(-1)^{i+j}M_{ij}). • Adjugate (adj(A)): transpose of cofactor matrix. • (A^{-1} = \dfrac{1}{\det A},adj(A)) (only if (\det A \neq 0)). |
| Rank | • Row‑Echelon Form (REF) or Reduced Row‑Echelon Form (RREF): number of non‑zero rows = rank. • Equivalent: maximum order of non‑zero minors. • Rank tells whether (AX=B) has unique solution, infinite solutions, or none. |
| Solving Linear Systems | • Gaussian Elimination: forward elimination to REF, then back‑substitution. • Cramer’s Rule (only for square, (\det A \neq 0)): (x_i = \dfrac{\det A_i}{\det A}) where (A_i) is (A) with its (i^{th}) column replaced by (B). |
| Determinant Properties | • (\det(AB) = \det A \cdot \det B). • (\det A^T = \det A). • (\det A = 0) iff (A) is singular (no inverse). • (\det(kA) = k^n \det A) for (n \times n) matrix. |
3. Worked Example
Problem
Solve the linear system
[
\begin{cases}
2x + 3y - z = 4\
x - y + 4z = 2\
3x + 2y + 5z = 7
\end{cases}
]
using Cramer’s Rule.
Step‑by‑Step
- Write the coefficient matrix (A) and vector (B).
[ A= \begin{bmatrix} 2 & 3 & -1\ 1 & -1 & 4\ 3 & 2 & 5 \end{bmatrix}, \qquad B= \begin{bmatrix} 4\ 2\ 7