Class 12 Maths and Statistics Part 1 Chapter 2: Matrices

Key concepts: Matrices

Matrix and order

A = [aᵢⱼ] with m rows and n columns has order m × n. aᵢⱼ is the element in row i and column j.

Types of matrices

Row, column, zero (null), square, diagonal, scalar, identity (unit) I, upper and lower triangular, symmetric (Aᵀ = A) and skew-symmetric (Aᵀ = −A, diagonal elements zero). |A| = 0: singular; |A| ≠ 0: non-singular.

Algebra of matrices

Addition and subtraction need the same order. kA = [kaᵢⱼ]. AB is defined when the number of columns of A equals the number of rows of B; an m × p matrix times a p × n matrix is m × n. In general AB ≠ BA.

Transpose

Aᵀ is obtained by interchanging rows and columns. (Aᵀ)ᵀ = A, (A + B)ᵀ = Aᵀ + Bᵀ, (kA)ᵀ = kAᵀ and (AB)ᵀ = BᵀAᵀ.

Symmetric + skew-symmetric

Every square matrix A = ½(A + Aᵀ) + ½(A − Aᵀ), a symmetric plus a skew-symmetric matrix.

Elementary transformations

Interchange two rows (Rᵢ ↔ Rⱼ), multiply a row by a non-zero scalar (kRᵢ) and add a multiple of one row to another (Rᵢ + kRⱼ); similarly for columns.

Inverse of a matrix

AA⁻¹ = A⁻¹A = I. A⁻¹ exists only if |A| ≠ 0. Adjoint method: A⁻¹ = (1/|A|) adj A, where adj A is the transpose of the cofactor matrix. For [a b; c d], A⁻¹ = (1/(ad − bc)) [d −b; −c a].

Solving linear equations

Write the system as AX = B. Method of inversion: X = A⁻¹B. Method of reduction: reduce A to upper triangular form by row transformations and solve by back substitution.

Quick revision: important points

• (AB)⁻¹ = B⁻¹A⁻¹ and (A⁻¹)ᵀ = (Aᵀ)⁻¹.
• A null square matrix is both symmetric and skew-symmetric.
• Use only row transformations with AA⁻¹ = I, or only column transformations with A⁻¹A = I.
• A(adj A) = (adj A)A = |A| I.
• Diagonal matrices are symmetric.

Practice MCQs with answers

1. A matrix with 3 rows and 2 columns has order:

   1. 2 × 3
   2. 3 × 2
   3. 6 × 1
   4. 3 × 3
   Show answer

   Answer: B. 3 × 2
   Order is rows × columns.

2. A square matrix in which all non-diagonal elements are zero is a:

   1. Null matrix
   2. Diagonal matrix
   3. Row matrix
   4. Skew-symmetric matrix
   Show answer

   Answer: B. Diagonal matrix
   Only diagonal elements may be non-zero.

3. A diagonal matrix with every diagonal element 1 is called:

   1. Scalar matrix of 0
   2. Identity matrix
   3. Null matrix
   4. Triangular matrix
   Show answer

   Answer: B. Identity matrix
   The unit or identity matrix I.

4. (Aᵀ)ᵀ equals:

   1. A
   2. −A
   3. A⁻¹
   4. I
   Show answer

   Answer: A. A
   Transposing twice returns the original matrix.

5. A square matrix A is symmetric if:

   1. Aᵀ = −A
   2. Aᵀ = A
   3. A² = A
   4. |A| = 0
   Show answer

   Answer: B. Aᵀ = A
   It equals its transpose.

All Class 12 Maths and Statistics chapters

• Part 1 Chapter 1: Mathematical Logic
• Part 1 Chapter 2: Matrices
• Part 1 Chapter 3: Differentiation
• Part 1 Chapter 4: Applications of Derivatives
• Part 1 Chapter 5: Integration
• Part 1 Chapter 6: Definite Integration
• Part 1 Chapter 7: Application of Definite Integration
• Part 1 Chapter 8: Differential Equation and Applications
• Part 2 Chapter 1: Commission, Brokerage and Discount
• Part 2 Chapter 2: Insurance and Annuity
• Part 2 Chapter 3: Linear Regression
• Part 2 Chapter 4: Time Series
• Part 2 Chapter 5: Index Numbers
• Part 2 Chapter 6: Linear Programming
• Part 2 Chapter 7: Assignment Problem and Sequencing
• Part 2 Chapter 8: Probability Distributions

Class 12 Maths and Statistics notes: all chapters