Question

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 \({I_n}\).

Short answer

1. False
2. True
3. True
4. True
5. False

Step-by-step solution

Use the fact of an inverse

(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.

Use the properties of inverse

(b)

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

Thus, the given statement is true.

Use the second statement of theorem 4

(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.

Use the first statement of theorem 7

(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.

Use the second statement of theorem 7

(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.