Question

TRUE OR FALSE

If ${\mathit{A}}^{\mathbf{T}}\mathit{A}\mathbf{=}\mathit{A}{\mathit{A}}^{\mathbf{T}}$or an $\mathit{n}\mathbf{\times }\mathit{n}$ matrix 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, 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 the 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,

$A·A^T=I$

Let A be an $n\times n$symmetricmatrix then,

$$\begin{gathered} A^T=A \\ \Rightarrow A^{T}A=A{A}^T \end{gathered}$$

But not every symmetric matrix is orthogonal. Being orthogonal means its transpose is its inverse, so it must be an invertible matrix and this is not necessary for a symmetric matrix.

Let’s take a $2\times 2$ matrix,

$$\begin{gathered} A=\left[12 \\ 24\right] \end{gathered}$$

Here $A^{T}A=A{A}^T$, but A is not an orthogonal matrix because $\det \left(A\right)=0$.

Hence, the given statement is false.