In Exercise 9 mark each statement True or False. Justify each answer.

9. a. In order for a matrix B to be the inverse of A, both equations \(AB = I\) and \(BA = I\) must be true.

b. If A and B are \(n \times n\) and invertible, then \({A^{ - {\bf{1}}}}{B^{ - {\bf{1}}}}\) is the inverse of \(AB\).

c. If \(A = \left( {\begin{aligned}{*{20}{c}}a&b\\c&d\end{aligned}} \right)\) and \(ab - cd \ne {\bf{0}}\), then A is invertible.

d. If A is an invertible \(n \times n\) matrix, then the equation \(Ax = b\) is consistent for each b in \({\mathbb{R}^{\bf{n}}}\).

e. Each elementary matrix is invertible.

(a)

Given that B is the inverse of A, which means A is invertible.

Hence, \(AB = BA = I\).

Since the inverse is uniquely determined by A, the given statement is true.

(b)

Note that AB is invertible. Then, \({\left( {AB} \right)^{ - 1}} = {B^{ - 1}}{A^{ - 1}}\).

Thus, the given statement is false.

(c)

Note that \(A = \left( {\begin{aligned}{*{20}{c}}a&b\\c&d\end{aligned}} \right)\) is invertible if and only if \(ad - bc \ne 0\).

Thus, the given statement is false.

(d)

If A is an invertible\(n \times n\)matrix, then for each bin\({\mathbb{R}^n}\), the equation\(Ax = b\)has the unique solution\(x = {A^{ - 1}}b\).

This implies that \(Ax = b\) is consistent.

Thus, the given statement is true.

(e)

The inverse exists for each elementary matrix. It means each elementary matrix is invertible.

Thus, the given statement is true.