Overview: Matrix | General Intelligence and Reasoning for SSC CGL PDF Download

Introduction

An m × n matrix is usually written as:

In brief, the above matrix is represented by A = [a_ij] _mxn. The numbers a₁₁, a₁₂, ….. etc., are known as the elements of the matrix A, where a_ij belongs to the i^th row and j^th column and is called the (i, j)^th element of the matrix A = [a_ij]. 

Important Formulas for Matrices 

If A and B are square matrices of order n, and I_n is a corresponding unit matrix, then

(a) A(adj.A) = | A | I_n = (adj A) A

(b) | adj A | = | A |n^-1 (Thus A (adj A) is always a scalar matrix)

(c) adj (adj.A) = | A |^n-2 A
(d) 

(f) adj (A^m) = (adj A)^m,

(g) adj(kA) = k^n-1 (adj.A), KϵR
(h) adj (I_n) = I_n 

(i) adj 0 = 0

(j) A is symmetric ⇒adj A is also symmetric

(k) A is diagonal ⇒adj A is also diagonal

(l) A is triangular ⇒adj A is also triangular

(m) A is singular ⇒| adj A | = 0

Types of Matrices

(i) Symmetric matrix: A square matrix A = [a_ij] is called a symmetric matrix if a_ij = a_ji, for all i, j.
(ii) Skew-symmetric matrix: when a_ij = – a_ji
(iii) Hermitian and skew – Hermitian matrix:

(Hermitian matrix)(A^θ represents conjugate transpose)

(skew-Hermitian matrix)
(iv) Orthogonal matrix: if AA^T = I_n = A^TA
(v) Idempotent matrix: if A² = A
(vi) Involuntary matrix: if A² = I or A^-1 = A
(vii) Nilpotent matrix: A square matrix A is nilpotent; if A^p = 0, p is an integer.

Trace of Matrix

The trace of a square matrix is the sum of the elements on the main diagonal.
(i) tr(λA_ = λ tr(A)
(ii) tr(A + B) = tr(A) + tr(B)
(iii) tr(AB) = tr(BA)

Matrix Transpose

Properties of Matrix Multiplication

(i) AB ≠ BA
(ii) (AB)C = A(BC)
(iii) A.(B + C) = A.B + A.C

Adjoint of a Matrix

Inverse of a Matrix

Order of a Matrix

A matrix which has m rows and n columns is called a matrix of order m x n.
For example, the order of

matrix is 2 x 3.

Note: (a) The matrix is just an arrangement of certain quantities.
(b) The elements of a matrix may be real or complex numbers. If all the elements of a matrix are real, then the matrix is called a real matrix.
(c) An m x n matrix has m.n elements.

Illustration 1: Construct a 3×4 matrix A = [a_ij], whose elements are given by a_ij = 2i + 3j.
Sol: In this problem, I and j are the number of rows and columns, respectively. By substituting the respective values of rows and columns in a_ij = 2i + 3j, we can construct the required matrix.
Given a_ij = 2i + 3j
so a₁₁ = 2+3 = 5, a₁₂ = 2+6 = 8
so a₁₁ = 2+3 = 5, a₁₂ = 2+6 = 8

Similarly, a₁₃ = 11, a₁₄=14, a₂₁ = 7, a₂₂=10, a₂₃=13, a₂₄=16,a₃₁=9, a₃₂=12, a₃₃=15, a₃₄=18

Illustration 2: Construct a 3 x 4 matrix, whose elements are given by: aij =
Sol: The method for solving this problem is the same as in the above problem.
Since


Hence, the required matrix is given by

Trace of a Matrix

Let A = [a_ij]_nxn and B = [b_ij]_nxn and λ be a scalar,
(i) tr(λA) = λ tr(A) (ii) tr(A + B) = tr(A) + tr(B) (iii) tr(AB) = tr(BA)

Transpose of Matrix

The matrix obtained from a given matrix A by changing its rows into columns or columns into rows is called the transpose of matrix A and is denoted by AT or A’. From the definition, it is obvious that if the order of A is m x n, then the order of AT becomes n x m; For example, transpose of a matrix.

Properties of Transpose of Matrix

(i) (A^T)^T= A
(ii) (A + B)^T = A^T+ B^T
(iii) (AB)^T = B^TA^T
(iv) (kA)^T = k(A)^T
(v) (A₁A₂A₃ ……A_n-1A_n)^T =
(vi) I^T = I (vii) tr(A) = tr(A^T)

Problems on Matrices

Illustration 3: If . then prove that (AB)T = BTAT.

Solution: By obtaining the transpose of AB, i.e., (AB)^T and multiplying B^T and A^T, we can easily get the result.
Here, AB =



Illustration 4: If Then what is (B’)’A’ equal to?
Sol: In this problem, we use the properties of the transpose of a matrix to get the required result.
We have =