In Exercise 10 mark each statement True or False. Justify each answer.

10. a. A product of invertible \(n \times n\) matrices is invertible, and the inverse of the product of their inverses in the same order.

b. If A is invertible, then the inverse of \({A^{ - {\bf{1}}}}\) is A itself.

c. If \(A = \left( {\begin{aligned}{*{20}{c}}a&b\\c&d\end{aligned}} \right)\) and \(ad = bc\), then A is not invertible.

d. If A can be row reduced to the identity matrix, then A must be invertible.

e. If A is invertible, then elementary row operations that reduce A to the identity \({I_n}\) also reduce \({A^{ - {\bf{1}}}}\) to \({I_n}\).

(a)

Note that\({\left( {AB} \right)^{ - 1}} = {B^{ - 1}}{A^{ - 1}}\). This implies that the product of inverses is in reverse order.

Thus, the given statement is false.

(b)

If A is invertible, then \({\left( {{A^{ - 1}}} \right)^{ - 1}} = A\).

Thus, the given statement is true.

(c)

According to this statement, \(A = \left( {\begin{aligned}{*{20}{c}}a&b\\c&d\end{aligned}} \right)\) is not invertible if \(ad - bc = 0\), i.e., \(ad = bc\).

Thus, the given statement is true.

(d)

According to this statement, Ais invertible if and only if it is row equivalent to the identity matrix.

Thus, the given statement is true.

(e)

According to this statement, if A is invertible, then any sequence of elementary row operations that reduces A to \({I_n}\) also transforms \({I_n}\) into \({A^{ - 1}}\).

Thus, the given statement is false.