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

Introduction

An m × n matrix is usually written as:
Overview: Matrix | General Intelligence and Reasoning for SSC CGL
In brief, the above matrix is represented by A = [aij] mxn. The numbers a11, a12, ….. etc., are known as the elements of the matrix A, where aij belongs to the ith row and jth column and is called the (i, j)th element of the matrix A = [aij]. 

Overview: Matrix | General Intelligence and Reasoning for SSC CGL

Important Formulas for Matrices 

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

(a) A(adj.A) = | A | In = (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) Overview: Matrix | General Intelligence and Reasoning for SSC CGL

(e) adj (AB) = (adj B) (adj A)

(f) adj (Am) = (adj A)m,

(g) adj(kA) = kn-1 (adj.A), KϵR
(h) adj (In) = In 

(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 = [aij] is called a symmetric matrix if aij = aji, for all i, j.
(ii) Skew-symmetric matrix: when aij = – aji
(iii) Hermitian and skew – Hermitian matrix:
Overview: Matrix | General Intelligence and Reasoning for SSC CGL

(Hermitian matrix)(Aθ represents conjugate transpose)
Overview: Matrix | General Intelligence and Reasoning for SSC CGL
(skew-Hermitian matrix)
(iv) Orthogonal matrix: if AAT = In = ATA
(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= 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
Overview: Matrix | General Intelligence and Reasoning for SSC CGL


Overview: Matrix | General Intelligence and Reasoning for SSC CGL


Properties of Matrix Multiplication


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

Adjoint of a Matrix

Overview: Matrix | General Intelligence and Reasoning for SSC CGL

Inverse of a Matrix


A-1 exists if A is non-singular, i.e.,
Overview: Matrix | General Intelligence and Reasoning for SSC CGL


 

 

 


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

Overview: Matrix | General Intelligence and Reasoning for SSC CGL
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 = [aij], whose elements are given by aij = 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 aij = 2i + 3j, we can construct the required matrix.
Given aij = 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
Overview: Matrix | General Intelligence and Reasoning for SSC CGL

Illustration 2: Construct a 3 x 4 matrix, whose elements are given by: aij =Overview: Matrix | General Intelligence and Reasoning for SSC CGL
Sol: The method for solving this problem is the same as in the above problem.
Since
Overview: Matrix | General Intelligence and Reasoning for SSC CGL
Overview: Matrix | General Intelligence and Reasoning for SSC CGL
Hence, the required matrix is given by
Overview: Matrix | General Intelligence and Reasoning for SSC CGL

Trace of a Matrix

Let A = [aij]nxn and B = [bij]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)
Overview: Matrix | General Intelligence and Reasoning for SSC CGL

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.
Overview: Matrix | General Intelligence and Reasoning for SSC CGL

Properties of Transpose of Matrix

(i) (AT)T= A
(ii) (A + B)T = AT+ BT
(iii) (AB)T = BTAT
(iv) (kA)T = k(A)T
(v) (A1A2A3 ……An-1An)T =Overview: Matrix | General Intelligence and Reasoning for SSC CGL
(vi) IT = I (vii) tr(A) = tr(AT)

Problems on Matrices

Illustration 3: If Overview: Matrix | General Intelligence and Reasoning for SSC CGL. then prove that (AB)T = BTAT.

Solution: By obtaining the transpose of AB, i.e., (AB)T and multiplying BT and AT, we can easily get the result.
Here, AB =
Overview: Matrix | General Intelligence and Reasoning for SSC CGL
Overview: Matrix | General Intelligence and Reasoning for SSC CGL

Illustration 4: IfOverview: Matrix | General Intelligence and Reasoning for SSC CGL 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 =
Overview: Matrix | General Intelligence and Reasoning for SSC CGL

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FAQs on Overview: Matrix - General Intelligence and Reasoning for SSC CGL

1. What is a matrix?
A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is denoted using brackets and its size is determined by the number of rows and columns it contains.
2. How do you add two matrices together?
To add two matrices, they must have the same dimensions. You simply add the corresponding elements of the matrices together. For example, if matrix A has elements aij and matrix B has elements bij, then the sum of matrix A and matrix B, denoted as A + B, will have elements aij + bij.
3. What is the determinant of a matrix?
The determinant of a square matrix is a scalar value that is computed using a specific formula. It provides information about the matrix, such as whether it is invertible or singular. The determinant of a 2x2 matrix [a b; c d] is given by the formula ad - bc.
4. How do you multiply two matrices together?
To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The product of matrix A and matrix B, denoted as AB, is obtained by multiplying the corresponding elements of the matrices and summing them up. The element at row i and column j of the resulting matrix is obtained by multiplying the elements of row i of matrix A with the corresponding elements of column j of matrix B and adding them.
5. What is the inverse of a matrix?
The inverse of a square matrix A is another matrix, denoted as A^(-1), such that when multiplied with A, it results in the identity matrix. The inverse of a matrix exists only if its determinant is non-zero. It is used to solve linear equations and various other mathematical operations.
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