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Chapter 7: Q29E (page 395)

Suppose A is invertible and orthogonally diagonalizable. Explain why \({A^{ - {\bf{1}}}}\) is also orthogonally diagonalizable.

Short Answer

Expert verified

The matrix \({A^{ - 1}}\) is diagonizable.

Step by step solution

01

Write the propertyofthe diagonizable matrix

As A is diagonizable, so for the matrices P and D, \(A = PD{P^{ - 1}}\).

02

Find the inverse of A

Apply the properties of the inverse \(A = PD{P^{ - 1}}\).

\(\begin{aligned}{}{A^{ - 1}} &= {\left( {PD{P^{ - 1}}} \right)^{ - 1}}\\ &= {\left( {{P^{ - 1}}} \right)^{ - 1}}{D^{ - 1}}{P^{ - 1}}\\ &= P{D^{ - 1}}{P^{ - 1}}\end{aligned}\)

Thus, the above equation shows that the matrix \({A^{ - 1}}\) is diagonizable.

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