IIM Bangalore (UGAT) Modern Math > Matrices & Determinants PYQs

Let $b$ and $c$ be two real numbers such that the numbers $3, b, c$ are in a geometric progression. For the matrix $A = \begin{bmatrix} 1 & 1 & 1 \\ 3 & b & c \\ 9 & b^2 & c^2 \end{bmatrix}$, if $\det[A] = 162$, then the value of $c$ is

Let $A = \begin{bmatrix} x & 1 \\ 0 & 1 \end{bmatrix}$ where $x$ is a real number, and $B = \begin{bmatrix} 2\sqrt{2} & 3 + \sqrt{2} \\ 0 & 1 \end{bmatrix}$. If $A^3 = B$, then $x$ is equal to

Let $A = [a_{ij}]$ be a 3 × 3 matrix such that $a_{ij} = i + j$, for all $i, j$. If $A^3 + kA^2 + lA + mI = 0$, then $k + l + m$ is equal to