Question

Give an example of a matrix which is its own inverse (that is, where A^-1 = A)

Answer 1

the identity matrix
1  0
0  1

Answer 2

Matrices are used to represent data in rows and columns

A matrix which is its own inverse is the identity matrix.

An 2-by-2 identity matrix is represented as:

$\mathbf{A = \left[\begin{array}{cc}1&0\\0&1\end{array}\right] }$

Calculate the determinant (d)

$\mathbf{d = 1 \times 1 - 0 \times 0}$

$\mathbf{d = 1}$

So, the inverse matrix is:

$\mathbf{A^{-1}= \frac{1}{d} A^T}$

This gives

$\mathbf{A^{-1}= \frac{1}{1} A^T}$

$\mathbf{A^{-1}= A^T}$

Transpose the matrix.

So, we have

$\mathbf{A^T = \left[\begin{array}{cc}1&0\\0&1\end{array}\right] }$

By comparison

$\mathbf{A=A^T = \left[\begin{array}{cc}1&0\\0&1\end{array}\right] }$

Hence, a matrix which is its own inverse is the identity matrix.

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