Matrices Formula: Definition, Properties, Types, Solution, Examples

A matrix is a structured arrangement of numbers, which can be either real or complex, organized in a rectangular grid format consisting of m horizontal lines (known as rows) and n vertical lines (known as columns). This grid is referred to as a matrix of order m by n, often denoted as an m × n matrix. The numbers within the matrix are enclosed within square brackets [ ] or parentheses ( ).

Introduction to Matrices

In a nutshell, the matrix described above can be represented by the notation A = [a _ij ] mxn. Here, the individual numbers a11, a12, and so on, are referred to as the elements of the matrix A. Each element, denoted as a _ij , belongs to the ith row and jth column, making it the (i, j)th element of the matrix A, which is written as A = [a _ij ].

CBSE Class 10 Sample Paper

Important Formulas for Matrices

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

• A(adj.A) = | A | In = (adj A) A
• | adj A | = |A| ^n-1 (Thus A (adj A) is always a scalar matrix)
• adj (adj.A) = | A | ^n-2 A
• adj (AB) = (adj B) (adj A)
• adj (Am) = (adj A)m,
• adj 0 = 0
• A is symmetric ⇒ adj A is also symmetric
• A is diagonal ⇒ adj A is also diagonal
• A is triangular ⇒ adj A is also triangular
• 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:

(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 Ap = 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)

Also Check – Comparing Quantities Formula

Matrix Transpose

• (AT)T= A
• (A+B)T= AT + BT
• (AB)T= BTAT
• (kA) ^T = k(A) ^T
• (A1A2A3…..An-1An) ^T = A ^T _n A ^t _n-1 …..A ^T ₃ A ^T ₂ A ^T ₁
• I ^T = I
• tr(A) = t(A ^T )

Properties of Matrix Multiplication

• AB ≠ BA
• (AB)C = A(BC)
• A (B + C) = AB + AC
• (A + B)C = AC + BC

Adjoint of a Matrix

Also Check – Congruence of Triangles

Inverse of a Matrix

A-1 exists if A is non-singular, i.e.,

Order of a Matrix

A matrix with m rows and n columns is designated as a matrix of order m x n.

For instance, consider the following matrix:

Please note the following:

(a) A matrix is simply a structured arrangement of specific quantities.

(b) The elements within a matrix can consist of real or complex numbers. If all the elements in a matrix are real numbers, it is referred to as a real matrix.

(c) An m x n matrix contains a total of m * n elements.

Illustration 1: Create a 3×4 matrix A = [aij] with elements defined as a _ij = 2i + 3j.

In this illustration, we will build a 3×4 matrix denoted as A, where each element a _ij is determined using the formula aij = 2i + 3j.

Solution: In this particular problem, we have 'i' representing the number of rows, and 'j' representing the number of columns. By substituting the appropriate values for rows and columns into the formula a _ij = 2i + 3j, we can construct the desired matrix.

Using the formula a _ij = 2i + 3j:

For the 3×4 matrix A, we can compute its elements as follows:

a11 = 2(1) + 3(1) = 5

a12 = 2(1) + 3(2) = 8

a13 = 2(1) + 3(3) = 11

a14 = 2(1) + 3(4) = 14

a21 = 2(2) + 3(1) = 7

a22 = 2(2) + 3(2) = 10

a23 = 2(2) + 3(3) = 13

a24 = 2(2) + 3(4) = 16

a31 = 2(3) + 3(1) = 9

a32 = 2(3) + 3(2) = 12

a33 = 2(3) + 3(3) = 15

a34 = 2(3) + 3(4) = 18

Using these calculations, we can construct the 3×4 matrix A with the respective values for its elements.

Also Check – Cubes and Cubes Roots Formula

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)

Transpose of Matrix

The transformation of a given matrix A by interchanging its rows and columns results in a new matrix known as the transpose of matrix A, denoted by AT or A'. According to this definition, if the original matrix A has an order of m x n, then the order of its transpose, AT, becomes n x m.

For instance, consider the transpose of a matrix:

Also Check – Factorization Formula

Properties of Transpose of Matrix

• (AT)T= A
• (ii) (A + B)T = AT+ BT
• (iii) (AB)T = BTAT
• (iv) (kA)T = k(A)T
• IT = I
• (vii) tr(A) = tr(AT)