Question

Which of the following statements is true regarding the invertibility of similar matrices?

A) If a matrix A is invertible, all matrices similar to A are also invertible.
B) If a matrix A is invertible, only some matrices similar to A are invertible.
C) If a matrix A is invertible, no matrices similar to A are invertible.
D) The invertibility of a matrix is not related to the invertibility of matrices similar to it.

Answer 1

Similar matrices share determinant values. Since a non-zero determinant indicates invertibility, if matrix A is invertible, all matrices similar to A are also invertible. The correct answer is option A.

Step-by-step explanation:

The question pertains to the invertibility of similar matrices. Two matrices A and B are said to be similar if there exists an invertible matrix P such that B = P⁻¹AP. An important property of similar matrices is that they share many characteristics, including their determinants.

Since the determinant of a matrix determines whether or not it is invertible (a matrix is invertible if and only if its determinant is non-zero), and since similar matrices have the same determinant, it follows that if a matrix A is invertible, then any matrix B that is similar to A must also be invertible.

Therefore, the correct statement regarding the invertibility of similar matrices is:

A) If a matrix A is invertible, all matrices similar to A are also invertible.