Verify the statements in Exercises 19-24 . The matrices are...

Verify the statements in Exercises $19-24 .$ The matrices are square. If $A$ is invertible and similar to $B,$ then $B$ is invertible and $A^{-1}$ is similar to $B^{-1} .\left[\text { Hint } : P^{-1} A P=B \text { for some invertible }\right.$ $P .$ Explain why $B$ is invertible. Then find an invertible $Q$ such that $Q^{-1} A^{-1} Q=B^{-1} \cdot ]$

Verify the statements in Exercises $19-24 .$ The matrices are square.
If $A$ is invertible and similar to $B,$ then $B$ is invertible and
$A^{-1}$ is similar to $B^{-1} .\left[\text { Hint } : P^{-1} A P=B \text { for some invertible }\right.$ $P .$ Explain why $B$ is invertible. Then find an invertible $Q$ such that $Q^{-1} A^{-1} Q=B^{-1} \cdot ]$

Key Concepts

Invertible Matrices

An invertible matrix is one that has a corresponding matrix, called its inverse, such that the product of the matrix and its inverse yields the identity matrix. This concept is central in linear algebra because it guarantees that a system of linear equations has a unique solution. Moreover, the existence of an inverse matrix allows one to 'reverse' the effect of a matrix transformation.

Matrix Similarity

Matrix similarity is a relation between two square matrices where one can be transformed into the other via a change of basis. Specifically, matrices A and B are similar if there exists an invertible matrix P such that P?¹AP = B. Similar matrices share many key properties, including eigenvalues, determinants, traces, and rank, which means they essentially represent the same linear transformation under different coordinate systems.

Similar Inverses

The property of similar matrices extends to their inverses. If an invertible matrix A is similar to matrix B, then B must also be invertible and the inverse of A is similar to the inverse of B. This follows from the relation P?¹AP = B, which implies that by appropriately manipulating the equation and using the fact that inverses reverse the effect of the original transformation, one constructs an invertible matrix Q such that Q?¹A?¹Q = B?¹. This highlights that similarity preserves the algebraic structure under inversion.

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