Class 12 Maths Revision Notes for Matrices of Chapter 3

CBSE Class 12 Mathematics Chapter 3 Matrices

• A matrix is an ordered rectangular array of numbers, real or complex or functions.
• A matrix having m rows and n columns is called a matrix of order m × n.
• Column matrix: A matrix with one column is denoted by .
• Row matrix: A matrix with one row is denoted by .
• Square matrix: An m × n matrix is a square matrix if m = n.
• Diagonal matrix: A =  is a diagonal matrix if  =0, when 
• Scalar matrix:  is a scalar matrix if    when  i ≠  j,     (k is some constant), when I=j.
• Identity matrix:  is an identity matrix, if 
• Zero matrix: A zero matrix has all its elements as zero.
• Equality of two matrices:   if (i) A and B are of same order, (ii) for all possible values of 
• Scalar multiplication:  

Also       – A = (–1)A

• A – B = A + (–1) B

A + B = B + A
(A + B) + C = A + (B + C), where A =  B =  and C =  are of same order.

• k(A + B) = kA + kB, where A and B are of same order, k is constant.
• (k + l ) A = kA + lA, where k and l are constant.
• where

• (i)      A(BC) = (AB)C,

(ii)     A(B + C) = AB + AC,
(iii)    (A + B)C = AC + BC

• 
• (i) (A′)′ = A, ·   (ii) (kA)′ = kA′, ·   (iii) (A + B)′ = A′ + B′, ·   (iv) (AB)′ = B′A′
• Symmetric matrix: A is a symmetric matrix if A′ = A.
• Skew-aymmetric matrix: A is a skew symmetric matrix if A′ = –A.
• Any square matrix can be represented as the sum of a symmetric and a skew symmetric matrix. In fact, A = (A + A’) + (A – A’), where (A + A’) is a symmetric matrix and (A – A’) is a skew-symmetric matrix.
• Equivalent matrices: Two matrices A and B are equivalent that is, A – B is A is obtained from the other by a sequence of elementary operations. Elementary operations of a matrix are as follows:

(i)      or  (interchange rows or columns)
(ii)     or 
(iii)    or 

• If A and B are two square matrices such that AB = BA = I, then B is the inverse matrix of A and is denoted by and A is the inverse of  B.
• Inverse of a square matrix, if it exists, is unique.