Matrices | Engineering Mathematics - Civil Engineering (CE) PDF Download

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= [a_ij] mxn , then A^T = [b_ij] nxm where b_ij = a_ji

➢ Properties of Transpose of a Matrix

• (A^T)^TT = A
• (A+B)^TT = A^TT + B^TT
• (AB)^TT = B^TTA^TT

➢ 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. (A^T) = 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 AA^T = A^TA = 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

➢ Properties of Adjoint

• A(Adj A) = (Adj A) A = |A| I_n
• Adj(AB) = (Adj B).(Adj A) 

➢ 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: