Question

TRUE OR FALSE

If Ais an invertible matrix such that ${\mathit{A}}^{\mathbf{-}\mathbf{1}}\mathbf{=}\mathit{A}$, then Amust be orthogonal.

Short answer

The given statement is false.

Step-by-step solution

Definition of an orthogonal matrix

A square matrix with real numbers or elements is said to be an orthogonal matrix if its transpose is equal to its inverse matrix.

Or it can be said that, when the product of a square matrix and its transpose gives an identity matrix, then the square matrix is known as an orthogonal matrix.

Consider an example

For example,

Suppose A is a square matrix with real elements and of$n\times n$order and$A^T$ is the transpose of A. Then according to the definition, if,$A^T=A^{-1}$is satisfied, then,

$A·A^T=I$

If A is an invertible matrix such that $A^{-1}=A$, then$A^2=I$.

This means that A is an involuntary matrix.

Consider the matrix

$$\begin{gathered} A=\left[3-2 \\ 4-3\right]\Rightarrow A^2=\left[10 \\ 01\right] \end{gathered}$$

Because its column vectors are not unit vectors. Thus, it can’t be an orthogonal matrix.

Hence, the statement is false.