1. True or False? Give convincing reasons for your answers.

(a) If $A + I$ is invertible then $-1$ is not an eigenvalue of $A$.

(a) If $A + I$ is invertible then $-1$ is not an eigenvalue of $A$.

(b) If $\det(A - A^2) = 0$, then either $0$ or $1$ is an eigenvalue of $A$.

(c) A matrix is invertible if and only if $0$ is not one of its eigenvalues.

(d) Similar matrices must have same determinants.

(e) Similar matrices must have same eigenvalues.

(f) The matrices $A = \left[ \begin{array}{c c} 2 & 1 \\ 1 & 2 \end{array} \right]$ and $B = \left[ \begin{array}{c c} 4 & 3 \\ -1 & 0 \end{array} \right]$ are similar.

(g) The matrices $A = \left[ \begin{array}{c c} -3 & 1 \\ 3 & 2 \end{array} \right]$ and $B = \left[ \begin{array}{c c} 1 & 2 \\ -1 & 0 \end{array} \right]$ are similar.

(h) A vector $X$ is called an eigenvector of a matrix $A$ if there is a number $λ$ such that $AX = λX$.

(i) The eigenvalues and eigenvectors of $A$ and $A^T$ are always the same.

(j) If two matrices have the same eigenvalues then they are similar.

(k) If a $4 \times 4$ matrix is diagonalizable, then it must have distinct eigenvalues.

(l) If a $3 \times 3$ matrix is not diagonalizable, then it must have an eigenvalue of multiplicity $3$.

2. Consider the matrix $A = \left[ \begin{array}{c c c} -2 & 1 & 1 \\ 1 & -2 & 1 \\ 1 & 1 & -2 \end{array} \right]$.

(a) Find the eigenvalues of $A$. (Note: Write the multiplicity of each eigenvalue)

(a) Find the eigenvalues of $A$. (Note: Write the multiplicity of each eigenvalue)

(b) Find an eigenbasis for each eigenspace.

(c) Is $A$ diagonalizable? If yes, diagonalize it.

(d) Calculate $A^6$ using the previous part.

3. Calculate $B^{12}$ for $B = \left[ \begin{array}{c c} 3 & 5 \\ -1 & -1 \end{array} \right]$.

Suppose $A, B$, and $C$ are three square matrices such that $A ~ B$ and $B ~ C$. Show that $A ~ C$.

5. Suppose an $n \times n$ matrix $A$ has $n$ distinct eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$. Show that $\det A = \lambda_1 \times \lambda_2 \times \cdots \times \lambda_n$.

Note: This is always true, even if the eigenvalues are not distinct. But you do not have to show it here.

Note: This is always true, even if the eigenvalues are not distinct. But you do not have to show it here.

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