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Chapter 2: Q10SE (page 93)

Suppose A is invertible. Explain why \({A^T}A\) is also invertible. Then show that \({A^{ - {\bf{1}}}} = {\left( {{A^T}A} \right)^{ - {\bf{1}}}}{A^T}\).

Short Answer

Expert verified

Note that the product of invertible matrices is invertible. That is why \({A^T}A\) is also invertible. Hence, \({A^{ - 1}} = {\left( {{A^T}A} \right)^{ - 1}}{A^T}\) is proved.

Step by step solution

01

Write the given data

Given, A is invertible.

02

Use the inverse matrix theorem

Thus, \({A^T}\) is also invertible as per statements (a) and (l) in the inverse matrix theorem. Note that the product of invertible matrices is invertible. That is why \({A^T}A\) is also invertible.

03

Use the inverse property

\(\begin{aligned}{c}{\left( {{A^T}A} \right)^{ - 1}}{A^T} = {A^{ - 1}}{\left( {{A^T}} \right)^{ - 1}}{A^T}\\ = {A^{ - 1}}I\\{\left( {{A^T}A} \right)^{ - 1}}{A^T} = {A^{ - 1}}\end{aligned}\)

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