Question

For the matrix \[ [A] = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 2 & 1 \\ 3 & 1 & 2 \end{bmatrix} \] which of the following statements is/are TRUE? 

• A. The eigenvalues of $[A]^T$ are the same as the eigenvalues of $[A]$
• B. The eigenvalues of $[A]^{-1}$ are the reciprocals of the eigenvalues of $[A]$
• C. The eigenvectors of $[A]^T$ are the same as the eigenvectors of $[A]$
• D. The eigenvectors of $[A]^{-1}$ are the same as the eigenvectors of $[A]$

Hint:

The eigenvalues of a matrix and its transpose are identical, as are the eigenvectors of a matrix and its inverse. For inverses, the eigenvalues are reciprocals.

Answer 1

The Correct Option is A, B, D

Given the matrix $[A]$, we need to analyze the provided statements:

Step 1: Eigenvalues of $[A]^T$ and $[A]$.
For any square matrix $[A]$, the eigenvalues of $[A]$ and $[A]^T$ are always the same. This is because the characteristic equation for both matrices is the same. Therefore, statement (A) is true.

Step 2: Eigenvalues of $[A]^{-1$.}
The eigenvalues of $[A]^{-1}$ are the reciprocals of the eigenvalues of $[A]$. This is a well-known property of matrix inversion, making statement (B) true.

Step 3: Eigenvectors of $[A]^T$ and $[A]$.
The eigenvectors of $[A]^T$ and $[A]$ are the same for a square matrix, as transposing the matrix does not affect the eigenvectors. Thus, statement (C) is false, and the eigenvectors are the same as those of $[A]$.

Step 4: Eigenvectors of $[A]^{-1$ and $[A]$.}
The eigenvectors of $[A]^{-1}$ are the same as those of $[A]$, but the eigenvalues are the reciprocals. Thus, statement (D) is true. \[ \boxed{\text{The correct answers are (A), (B), and (D).}} \]