Matrices Set-2

Q1. Let $A$ be a $2\times2$ real matrix with $\operatorname{tr}(A)=5$ and $\det(A)=6$. The eigenvalues of $A$ are:

Q2. Let $A$ be a $2\times2$ real matrix with $\operatorname{tr}(A)=5$ and $\det(A)=3$. Let $M$ be the $4\times4$ block matrix $M=\begin{pmatrix}A & I\\ I & A\end{pmatrix}$ where $I$ is the $2\times2$ identity matrix. Then $\det(M)=$

Q3. Let $A$ be a real $3\times3$ orthogonal matrix ($A^T A = I$) with $\operatorname{tr}(A)=-1$. Which of the following must hold?

Q4. Let $A$ be a $3\times3$ real matrix such that $\operatorname{adj}(A)=A^T$. Then which of the following must hold for $\det(A)$?

Q5. Let $A$ and $B$ be $2\times2$ real matrices. Which of the following statements about the equation $AB-BA=I$ is correct?

Q6. Let $A$ be a $3\times3$ real matrix satisfying $A^2=2A$ and $\operatorname{rank}(A)=1$. Then $\operatorname{tr}(A)=$?

Q7. For which value of $k$ does the system

have infinitely many solutions?

Q8. Let $A$ be a $3\times3$ real matrix satisfying $A^2-5A+6I_3=0$ and $\operatorname{rank}(A-2I_3)=1$. Then $\operatorname{rank}(A-3I_3)=$?

Q9. If $A$ is a $3\times3$ matrix with eigenvalues $1,2,3$, then $\det\big(\operatorname{adj}(A)+2I_3\big)=$?

Q10. Let $A$ be a $3\times3$ real matrix with $\operatorname{tr}(A)=5$ and $\operatorname{tr}(A^2)=35$. Then $\operatorname{tr}\big(\operatorname{adj}(A)\big)=$?