Question

If is an invertible $\mathit{n}\mathbf{\times }\mathit{n}$matrix, then must commute with its adjoint, adj(A) .

Short answer

$A·adjA=adjA·A$

So, the given statement is true.

Step-by-step solution

Matrix Definition

Matrix is aset of numbers arranged in rows and columns so as to form a rectangulararray.

The numbers are called the elements, or entries, of the matrix.

If there are m rows and n columns, the matrix is said to be a “ m by n” matrix, written “ $m\times n$”.

To check whether the given condition is true or false

Ifis an invertible matrix, then

$A^{-1}=\frac{1}{\det A}adjA$.

Then,

$$\begin{gathered} A·adjA=A·\left(\det A\right)A^{-1} \\ A·adjA=\left(\det A\right)A{A}^{-1} \\ A·adjA=\left(\det A\right) \\ A·adjA=\left(\det A\right)A^{-1}A \end{gathered}$$

Therefore,

$A·adjA=adjA·A$