📐Mathematics
Matrices & Determinants
Types of matrices, operations, inverse, determinants, Cramer's rule, and system of equations.
Class 10Class 11Class 12
What You Will Learn in Matrices & Determinants
Matrices are rectangular arrays of numbers with operations (addition, multiplication, inverse), and determinants are scalar values computed from square matrices encoding geometric/algebraic properties.
- Matrix types: square, rectangular, diagonal, identity (I), zero, symmetric (A=Aᵀ), skew-symmetric.
- Multiplication: (A×B)ᵢⱼ = Σ AᵢₖBₖⱼ; non-commutative; requires cols(A)=rows(B).
- Determinant (2×2): |[a,b;c,d]| = ad-bc. (3×3): expansion along any row/column.
- Inverse: A⁻¹ = adj(A)/det(A); exists only if det(A)≠0 (non-singular matrix).
- Cramer's rule: for AX=B, xᵢ = det(Aᵢ)/det(A) where Aᵢ has B replacing ith column.
- Properties of determinants: |AB|=|A||B|; |kA|=kⁿ|A|; |Aᵀ|=|A|; row ops affect determinant.
Key Formulas
|A|₂ₓ₂ = ad-bcA⁻¹ = (1/|A|)·adj(A)Area of triangle = ½|x₁(y₂-y₃)+x₂(y₃-y₁)+x₃(y₁-y₂)|
Example
For A=[[2,3],[1,4]], det(A)=8-3=5, A⁻¹=(1/5)×[[4,-3],[-1,2]].
Expected Exam Questions — Matrices & Determinants
Q1.Find the inverse of matrix A = [[1,2],[3,4]].
Answer: det(A) = 1×4-2×3 = -2. adj(A) = [[4,-2],[-3,1]]. A⁻¹ = adj(A)/det(A) = (1/-2)[[4,-2],[-3,1]] = [[-2,1],[3/2,-1/2]].
Q2.If A = [[2,0],[0,3]], find |A⁵|.
Answer: |A| = 2×3 = 6. |A⁵| = |A|⁵ = 6⁵ = 7776.
Q3.For a 3×3 matrix, if two rows are identical, what is its determinant?
Answer: det = 0. Property: if any two rows (or columns) are identical, the determinant is zero. This means the matrix is singular (non-invertible).
🔘 MCQ Practice — Matrices & Determinants
MCQ 1.If A is a 3×3 matrix with det(A) = 5, then det(2A) is:
A. 10
B. 40 ✓
C. 25
D. 8
✓ Correct Answer: 40
Download Matrices & Determinants PDF Notes
Get the complete Matrices & Determinants notes as a PDF — free for enrolled students, or browse our public study materials library.