Matrices

Board: CBSE | Class: Class 12

Comprehensive study notes for Matrices by Ajay Yadav.

Key Concepts

Definition

A matrix is a rectangular array of numbers arranged in m rows and n columns, denoted as A = (a_ij)_m×_n. Order: m×n.

Types of Matrices

Row/Column: One row/column. Square: m=n. Diagonal: non-zero only on diagonal. Scalar: diagonal with all equal entries. Identity: diagonal = 1. Zero: all entries 0.

Operations

Addition: add corresponding entries. Scalar multiplication: multiply each entry. Multiplication: (AB)_ij = Σa_ikb_kj. AB exists if columns(A)=rows(B).

Transpose and Symmetry

(A’)_ij = A_ji. A is symmetric if A’=A. A is skew-symmetric if A’=-A (diagonal entries zero). Any square matrix = symmetric + skew-symmetric.

Elementary Operations

Row/column operations: R_i ↔ R_j, R_i → kR_i, R_i → R_i + kR_j. Used to find inverse and solve equations.

Important Formulas

Addition	A + B = (aij+bij)
Scalar mult	kA = (kaij)
Multiplication	(AB)ij = Σni(21 aikbkj
Transpose	(A')ij = Aji
Symmetric	A' = A
Skew-symmetric	A' = -A

Solved Examples

Example 1: Find AB if A=(1234), B=(5678).

Solution: AB=(1(5)+2(7)1(6)+2(8)3(5)+4(7)3(6)+4(8)) = (19224350).

Example 2: Show A=(1221) is symmetric.

Solution: A’=(1221)=A. Symmetric.

Example 3: Find X if 2X + (1234) = (5678).

Solution: 2X = (4444) ⇒ X = (2222).

Practice Questions

1. Find AB and BA for A=(1001), B=(2345). Is AB=BA?
2. Show any matrix can be expressed as sum of symmetric and skew-symmetric.
3. If A=(1234), find 2A-3I.
4. Find A’ for A=(123456).
5. For A=(1-123), find A².

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