Properties of Matrix Multiplication - Inverse of a Square Matrix by the Adjoint Method

If A is an invertible matrix, then det (A⁻¹) is equal to ____________ .

Find the inverse by using elementary row transformations:

\[\begin{bmatrix}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix}\]

Find the inverse by using elementary row transformations:

\[\begin{bmatrix}7 & 1 \\ 4 & - 3\end{bmatrix}\]

If \[A = \begin{bmatrix}- 1 & 2 & 0 \\ - 1 & 1 & 1 \\ 0 & 1 & 0\end{bmatrix}\] , show that  \[A^2 = A^{- 1} .\]

Find the adjoint of the following matrix:
\[\begin{bmatrix}a & b \\ c & d\end{bmatrix}\]

If A is an invertible matrix, then which of the following is not true ?

`("aA")^-1 = 1/"a"  "A"^-1`, where a is any real number and A is a square matrix.

If \[A = \begin{bmatrix}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{bmatrix}\] , then the value of |adj A| is _____________ .