Question

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.

Short answer

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

Step-by-step solution

Use the definition of 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.

Use the properties of an inverse

(b)

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

Thus, the given statement is false.

Use the definition of an invertible matrix

(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.

Use theorem 5

(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.

Use the fact of elementary matrix

(e)

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

Thus, the given statement is true.