Question

If a square matrix $\mathit{A}$ is invertible, then its classical adjoint $\mathbf{adj}\left(A\right)$ is invertible as well.

Short answer

Therefore, the $adj\left(A\right)$is invertible, $detadjA\ne 0$ So, the given statement is true.

Step-by-step solution

Orthogonal Matrix Definition

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 we can say, when the product of a square matrix and its transpose gives an identity matrix, then the square matrix is known as an orthogonal matrix.

To check whether the given condition is true or false

If a square matrix $A$is invertible, then we have

$A^{-1}=\frac{1}{detA}adjA$

Therefore,

$$\begin{gathered} 0\ne det{A}^{-1} \\ =det\left(\frac{1}{detA}adjA\right) \end{gathered}$$

$detadjA\ne 0$

Therefore, the $adj\left(A\right)$is invertible, $detadjA\ne 0$ So, the given statement is true.