# Matrices Notes | Study GATE Computer Science Engineering(CSE) 2023 Mock Test Series - Computer Science Engineering (CSE)

## Document Description: Matrices for Computer Science Engineering (CSE) 2022 is part of GATE Computer Science Engineering(CSE) 2023 Mock Test Series preparation. The notes and questions for Matrices have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about Matrices covers topics like Introduction and Matrices Example, for Computer Science Engineering (CSE) 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Matrices.

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Introduction

• A matrix represents a collection of numbers arranged in an order of rows and columns. It is necessary to enclose the elements of a matrix in parentheses or brackets.

A matrix with 9 elements is shown below:

This Matrix [M] has 3 rows and 3 columns. Each element of matrix [M] can be referred to by its row and column number.
Example: a23 = 6

➢ Order of a Matrix

• The order of a matrix is defined in terms of its number of rows and columns.
Order of a matrix = No. of rows ×No. of columns
• Therefore Matrix [M] is a matrix of order 3 × 3.

➢ Transpose of a Matrix

• The transpose [M]T of an m x n matrix [M] is the n x m matrix obtained by interchanging the rows and columns of [M].
• If A= [aij] mxn , then AT = [bij] nxm where bij = aji

➢ Properties of Transpose of a Matrix

• (AT)TT = A
• (A+B)TT = ATT + BTT
• (AB)TT = BTTATT

➢ Singular and Nonsingular Matrix

• Singular Matrix: A square matrix is said to be singular matrix if its determinant is zero i.e. |A|=0.
• Nonsingular Matrix: A square matrix is said to be non-singular matrix if its determinant is non-zero.

➢ Properties of Matrix Addition and Multiplication

• A+B = B+A (Commutative)
• (A+B)+C = A+(B+C) (Associative)
• AB ≠ BA (Not Commutative)
• (AB) C = A (BC) (Associative)
• A (B+C) = AB+AC (Distributive)

➢ Square Matrix

• A square Matrix has as many rows as it has columns. i.e. no of rows = no of columns.

➢ Symmetric Matrix

• A square matrix is said to be symmetric if the transpose of original matrix is equal to its original matrix. i.e. (AT) = A.

➢ Diagonal Matrix

• A Symmetric matrix is said to be diagonal matrix where all the off diagonal elements are 0.

➢ Identity Matrix

• A diagonal matrix with 1s and only 1s on the diagonal. Identity matrix is denoted as I.

➢ Orthogonal Matrix

• A matrix is said to be orthogonal if AAT = ATA = I

➢ Idempotent Matrix

• A matrix is said to be idempotent if A2 = A

➢ Involutory Matrix: A matrix is said to be Involutory if A2 = I.

Note: Every Square Matrix can uniquely be expressed as the sum of a symmetrix matrix and skew symmetric matrix. A = 1/2 (AT + A) + 1/2 (A – AT).

➢ Adjoint of a Square Matrix

➢ Inverse of a Square Matrix

• A-1 = Adj A / |A| ; |A|#0

➢ Properties of Inverse

• (A-1)-1 = A
• (AB)-1 = B-1A-1
• Only a non-singular square matrix can have an inverse.

➢ Where should we use Inverse Matrix?

• If you have a set of simultaneous equations:
► 7x + 2y + z = 21
►  3y – z = 5
► -3x + 4y – 2x = -1
• As we know when AX = B, then X = A-1B so we calculate the inverse of A and by multiplying it B, we can get the values of x, y and z.

➢ Trace of a Matrix

• Trace of a matrix is denoted as tr(A) which is used only for square matrix and equals the sum of the diagonal elements of the matrix.
Example:
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