# Matrices

## Involutory Matrix – Definition and Examples

Here you will learn what is involutory matrix with examples. Let’s begin – Involutory Matrix If $$A^2$$ = I . the matrix A is said to be an involutory matrix, i.e. the square roots of the identity matrix (I) is involutory matrix. Note : The determinant value of this matrix (A) is 1 or -1. …

## Idempotent Matrix – Definition and Example

Here you will learn what is idempotent matrix with examples. Let’s begin – Idempotent Matrix A square matrix is idempotent matrix provided $$A^2$$ = A. For this matrix note the following : (i) $$A^n$$ = A $$\forall$$ n $$\ge$$ 2, n $$\in$$ N. (ii) The determinant value of this matrix is either 1 or 0. …

## Formula for Inverse of a Matrix – Properties, Example

Here you will learn formula for inverse of a matrix and properties of inverse of matrix with example. Let’s begin – Formula for Inverse of a Matrix A square matrix A said to be invertible if and only if it is non-singular (i.e. |A| $$\ne$$ 0) and there exists a matrix B such that, AB …

## Adjoint of the Matrix (2×2 & 3×3) – Properties, Examples

Here you will learn how to find adjoint of the matrix 2×2 and 3×3, cofactors and its properties with examples. Let’s begin – Adjoint of the Matrix Let A = $$[a_{ij}]$$ be a square matrix of order n and let $$C_{ij}$$ be a cofactor of $$a_{ij}$$ in A. Then the transpose of the matrix of …

## How to Find Trace of Matrix – Properties & Example

Here you will learn how to find trace of matrix, its properties and what is orthogonal matrix with example. Let’s begin – Trace of Matrix The sum of the elements of the square matrix A lying along the principal diagonal is called the trace of A i.e (tr(A)).  Thus if  A = $$[a_{ij}]_{n\times n}$$, then …

## Properties of Multiplication of Matrices

Here you will learn properties of multiplication of matrices, positive integral powers of square matrix and matrix polynomial. Let’s begin – Properties of Multiplication of Matrices (a) Matrix multiplication is not commutative in general i.e AB $$\ne$$ BA. (b) Matrix multiplication is associative i.e. (AB) C = A (BC), whenever both sides are defined. (c) …