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Introduction of Matrix Notes | Study Engineering Mathematics - Civil Engineering (CE)

Document Description: Introduction of Matrix for Civil Engineering (CE) 2022 is part of Engineering Mathematics preparation. The notes and questions for Introduction of Matrix have been prepared according to the Civil Engineering (CE) exam syllabus. Information about Introduction of Matrix covers topics like Introduction and Introduction of Matrix Example, for Civil Engineering (CE) 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Introduction of Matrix.

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Introduction
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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.
Introduction of Matrix Notes | Study Engineering Mathematics - Civil Engineering (CE)
This Matrix [M] has 3 rows and 3 columns. Each element of matrix [M] can be referred to by its row and column number. For 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]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)T = A
  • (A + B)T = AT + BT
  • (AB)T = BTAT

Singular and Nonsingular Matrix 

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

Properties of Matrix addition and multiplication

  1. A + B = B + A (Commutative)
  2. (A + B) + C = A + (B + C) (Associative)
  3. AB ? BA (Not Commutative)
  4. (AB) C = A (BC) (Associative)
  5. 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. 

Skew-symmetric: A skew-symmetric (or antisymmetric or antimetric[1]) matrix is a square matrix whose transpose equals its negative.i.e. (AT) = -A. 

Diagonal Matrix: A diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The term usually refers to square matrices. 

Identity Matrix: A square matrix in which all the elements of the principal diagonal are ones and all other elements are zeros. Identity matrix is denoted as I. 

Orthogonal Matrix: A matrix is said to be orthogonal if AAT = ATA = I 

Idemponent Matrix: A matrix is said to be idemponent if A2 = A 

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

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

Adjoint of a square matrix: The adjoint of a matrix A is the transpose of the cofactor matrix of A

Introduction of Matrix Notes | Study Engineering Mathematics - Civil Engineering (CE)

Introduction of Matrix Notes | Study Engineering Mathematics - Civil Engineering (CE)

Introduction of Matrix Notes | Study Engineering Mathematics - Civil Engineering (CE)

Properties of Adjoint 

  1. A(Adj A) = (Adj A) A = |A| In
  2. Adj(AB) = (Adj B).(Adj A)
  3. |Adj A| = |A|n - 1
  4. Adj(kA) = kn - 1 Adj(A)
  5. |adj(adj(A))| = |A|^(n - 1)^2
  6. adj(adj(A)) = |A|^(n - 2)     *  A
  7. If A = [L, M, N] then adj(A) = [MN, LN, LM]
  8. adj(I) = I

Where, “n = number of rows = number of columns”

Inverse of a square matrix 

Introduction of Matrix Notes | Study Engineering Mathematics - Civil Engineering (CE)

Here |A| should not be equal to zero, means matrix A should be non-singular. 

Properties of inverse

  1. (A-1)-1 = A 
  2. (AB)-1 = B-1A-1 
  3. only a non singular square matrix can have an inverse. 

Where should we use the 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. Remember trace of a matrix is also equal to sum of eigen value of the matrix.
For example:

Introduction of Matrix Notes | Study Engineering Mathematics - Civil Engineering (CE)

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