Matrices Set-1

Q1. Let $A=\begin{pmatrix}2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2\end{pmatrix}$. Compute $\det(A)$.

Q2. Let $A$ be a real $3\times3$ matrix satisfying $A^2=3A-2I$. Which of the following could be $\det(A)$?

Q3. Let $A$ be a real $3\times3$ matrix of rank $1$ with $\operatorname{tr}(A)=5$. Find $\det(I+A)$.

Q4. Let $A$ be a real $2\times2$ matrix with $A^3=I$ and $A\ne I$. Then $\operatorname{tr}(A)$ equals:

Q5. For $t\in\mathbb{R}$ let $A(t)=\begin{pmatrix}t & 1 & 1 \\ 1 & t & 1 \\ 1 & 1 & t\end{pmatrix}$. For $t\ne1,-2$ the matrix is invertible. Find $t$ (with $t\ne1,-2$) such that $\operatorname{tr}\big(A(t)^{-1}\big)=0$.

Q6. Let

. Find

.

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

have infinitely many solutions?

Q8. Let $A$ be a $2\times2$ matrix with $\operatorname{tr}(A)=5$ and $\det(A)=6$. Then $\det(A-2I)$ equals

Q9. Let $A$ be an invertible $3\times3$ matrix and $u,v\in\mathbb{R}^3$ column vectors. If $\det(A)=2$ and $\det\bigl(A+uv^{T}\bigr)=10$, the scalar $v^{T}A^{-1}u$ equals

Q10. Let $A$ be a $3\times3$ matrix satisfying $\operatorname{adj}(A)=A$. Then $\det(A)$ is