# Involutory Matrix Calculator Online

### Explanation:

A matrix is said to be involutory if the square of the matrix is equal to identity matrix.
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In mathematics, there is a specific kind of matrix called an involutory matrix. An invertible matrix, or a non-singular square matrix with an existing inverse, is required for a matrix to be involutory. A square matrix that is involutory has a product with itself that equals the identity matrix of the same or identical order. An involutory matrix is, in other words, the inverse of itself. According to this, a matrix is said to be involutory if its square equals that of the identity matrix.

### What is An Involutory Matrix?

The self-inverse function is satisfied by an involutory matrix, which is its own inverse, making it a unique type of matrix. In simple English, it can be said that a square matrix A of order n is an involutory matrix if its square is equal or identical to the identity matrix of the same order. The square root of the identity matrix of order n is shared by all involutory matrices of order n.

### Involutory Matrix Properties

Since we now understand what the involutory matrix is, it will be easier to apply it and comprehend the idea. Let's go over some of its key characteristics.

• If the matrix A and matrix B are involutory matrices such that AB is equal to BA (we can say AB=BA), then AB is also an involutory matrix.
• A matrix "A" (or block diagonal Matrix A) that is derived from an involutory matrix is the matrix which is also an involutory matrix.
• Involutory matrices always have +1 and -1 as their eigenvalues.
• Any involutory matrix's determinant is always equal to +1 and -1.
• Each and every symmetric involutory matrix is an orthogonal involutory matrix, and the reverse is also true.
• A matrix An is also involutory for all integers n if A is involutory. If n is even, then An = I, and if n is odd, then An = A.
• If and only if A is an identity matrix then the involutory matrix A is an idempotent matrix
• A square matrix A is called as involutory if and only if it satisfies A2 = I or A = A-1.
• All involutory matrices having order n are square roots of identity matrix of order n.

#### How to use check Involutory Matrix Calculator?

• Firstly, you need to enter the dimension of the matrix. Enter number of rows in "Rows" input field and Enter number of columns in "Columns" input field.
• Then press the button "Set Matrix".
• An empty matrix will appear below and then you can enter your values inside the matrix.
• After entering all the values press "Solve" button, the result will automatically appear below which check whether the matrix is Involutory matrix or not.

Involutory Matrix Example Image:

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