Question

If A is similar to B, and A is invertible, then B must be invertible as well.

Short answer

The above statement is true.

If A is similar to B, and A is invertible, then B must be invertible as well.

Step-by-step solution

Definition of similar matrix

Let A and B are two square matrices, then the matrix A is said to be similar to the matrix B if there exists an invertible matrix P such that

$$\begin{gathered} A=P^{-1}BP \\ \det \left(A^{-1}\right)=\frac{1}{\det \left(A\right)}......\left(1\right) \end{gathered}$$

To show whether the matrix B is an invertible matrix or not

We know that if A and B are any two matrices of order n x n then

$\det \left(A^{-1}\right)=\frac{1}{\det \left(A\right)}$

And$\det \left(AB\right)=\det \left(A\right)\det \left(B\right)$

Using above properties in equation (1), we get

$$\begin{gathered} \det \left(A\right)=\det \left(P^{-1}AP\right) \\ \Rightarrow \det \left(A\right)=\det \left(P^{-1}\right)\det \left(A\right)\det \left(P\right) \end{gathered}$$

$\Rightarrow \det \left(A\right)=\frac{1}{\det \left(P\right)}\det \left(A\right)\det \left(B\right)$

$\Rightarrow \det \left(A\right)=\det \left(B\right)$

Since matrix A is invertible, therefore matrix B is also invertible.

Final Answer

If A is similar to B, and A is invertible, then B must be invertible as well since _{$\det \left(A\right)=\det \left(B\right).$}