Diagonalisation of matrices Quiz

1. Which matrix cannot be written as $UDU^{-1}$ with $U$ and $D$ real matrices and $D$ diagonal?

• A. $\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$
• B. $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$
• C. $\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}$
• D. $\begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}$

2. If $M=UDU^{-1}$, which expression gives $M^3$?

• A. $U^3DU^{-3}$
• B. $D^3$
• C. $UD^3U$
• D. $UD^3U^{-1}$

3. A matrix $M$ has eigenvalues $2$ and $3$. What are the eigenvalues of $M^5$?

• A. $32$ and $243$
• B. $10$ and $15$
• C. $7$ and $8$
• D. $5$ and $5$

4. In $M=UDU^{-1}$, the columns of $U$ are:

• A. the eigenvalues of $M$
• B. the rows of $D$
• C. the eigenvectors of $M$
• D. the entries of $M^2$

5. $M$ has eigenvalues $2$ and $3$. Find $\det(M^6)$.

• A. $36$
• B. $46656$
• C. $15625$
• D. $793$