Solved Examples on involutory Matrix

Example 1: Verify whether the matrix given below is involutory or not.

Solution:

To prove that the given matrix is involutory, we have to prove that A² = A.

Hence, verified.

So, the given matrix A is an involutory matrix.

Example 2: Give an example of an involutory matrix of order 2 × 2.

Solution:

A matrix  is said to be an involutory matrix, if a² + bc = 1.

Let us consider that a = 3, b = 4, c = −2 such that a² + bc = 1.

(3)² + (4) × (−2) = 9 − 8 = 1

We know that d = −a.

So, the involutory matrix is .

Example 3: Prove that the matrix given below is involutory.

Solution:

To prove that the given matrix is involutory, we have to prove that B = B^-1.

B^-1 = Adj B/ |B|

|B| = −49 − (−48) = −1

Hence, the given matrix is involutory.

Example 4: Prove that the determinant of an involutory matrix given below is always ±1.

Solution:

Let us consider of an involutory matrix “P” of order “n × n” to prove that its determinant is always ±1.

We know that a square matrix “P” is said to be involutory if and only if P² = I.

 P × P = I

Now, |P| × |P| = |I|

We know that the determinant of an identity matrix of any order is 1.

 (|P|)² = 1

 |P| = √1 = ±1

Thus, the determinant of an involutory matrix of any order is always ±1.

Hence proved.

Involutory Matrix is defined as the matrix that follows self inverse function i.e. the inverse of the Involutory matrix is the matrix itself. A matrix is defined as a rectangular array of numbers that are arranged in rows and columns. The size of a matrix can be determined by the number of rows and columns in it. A matrix is said to be an “m by n” matrix when it has “m” rows and “n” columns and is written as an “m × n” matrix. For example, a matrix of order “5 × 6” has five rows and six columns. We have different types of matrices, like rectangular, square, triangular, symmetric, singular, etc.