Properties of determinants; area of a triangle — practice questions

Question 1 · difficulty L2 · understanding

Let $$P = [{a_{ij}}]$$ be a 3 $$\times$$ 3 matrix and let $$Q = [{b_{ij}}]$$, where $${b_{ij}} = {2^{i + j}}{a_{ij}}$$ for $$1 \le i,j \le 3$$. If the determinant of P is 2, then the determinant of the matrix Q is

• A. 2 10
• B. 2 11
• C. 2 12
• D. 2 13

Question 2 · difficulty L3 · understanding

Let $|\mathrm{A}|=6$, where A is a $3 \times 3$ matrix. If $\left|\operatorname{adj}\left(3\operatorname{adj}\left(\mathrm{A}^2 \cdot \operatorname{adj}(2 \mathrm{~A})\right)\right)\right|=2^{\mathrm{m}} \cdot 3^{\mathrm{n}}, \mathrm{m}, \mathrm{n} \in \mathbf{N}$, then $\mathrm{m}+\mathrm{n}$ is equal to $\_\_\_\_$ .

Question 3 · difficulty L3 · understanding

Let $$A$$ be a $$3 \times 3$$ matrix of non-negative real elements such that $$A\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=3\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$$. Then the maximum value of $$\operatorname{det}(\mathrm{A})$$ is _________.

Question 4 · difficulty L3 · understanding

Let A be a $n \times n$ matrix such that $|\mathrm{A}|=2$. If the determinant of the matrix $\operatorname{Adj}\left(2 \cdot \operatorname{Adj}\left(2 \mathrm{~A}^{-1}\right)\right) \cdot$ is $2^{84}$, then $\mathrm{n}$ is equal to :

Question 5 · difficulty L3 · understanding

The positive value of the determinant of the matrix A, whose Adj(Adj(A)) = \begin{aligned}\left( {\matrix{ {14} & {28} & { - 14} \cr { - 14} & {14} & {28} \cr {28} & { - 14} & {14} \cr } } \right)\end{aligned}, is _____________.

Question 6 · difficulty L3 · understanding

Let A be a 3 $$\times$$ 3 real matrix. If det(2Adj(2 Adj(Adj(2A)))) = 2 41 , then the value of det(A 2 ) equal __________.

Question 7 · difficulty L3 · understanding

Let \begin{aligned}M = \left\{ {A = \left( {\matrix{ a & b \cr c & d \cr } } \right):a,b,c,d \in \{ \pm 3, \pm 2, \pm 1,0\} } \right\}\end{aligned}. Define f : M $$\to$$ Z, as f(A) = det(A), for all A$$\in$$M, where z is set of all integers. Then the number of A$$\in$$M such that f(A) = 15 is equal to _____________.

Question 8 · difficulty L3 · understanding

Let $$A = \{ {a_{ij}}\}$$ be a 3 $$\times$$ 3 matrix, where \begin{aligned}{a_{ij}} = \left\{ {\matrix{ {{{( - 1)}^{j - i}}} & {if} & {i < j,} \cr 2 & {if} & {i = j,} \cr {{{( - 1)}^{i + j}}} & {if} & {i > j} \cr } } \right.\end{aligned} then $$\det (3Adj(2{A^{ - 1}}))$$ is equal to _____________.

Question 9 · difficulty L3 · understanding

If \begin{aligned}A = \left[ {\matrix{ 2 & 3 \cr 0 & { - 1} \cr } } \right]\end{aligned}, then the value of det(A 4 ) + det(A 10 $$-$$ (Adj(2A)) 10 ) is equal to _____________.

Question 10 · difficulty L3 · understanding

Let \begin{aligned}P = \left[ {\matrix{ { - 30} & {20} & {56} \cr {90} & {140} & {112} \cr {120} & {60} & {14} \cr } } \right]\end{aligned} and \begin{aligned}A = \left[ {\matrix{ 2 & 7 & {{\omega ^2}} \cr { - 1} & { - \omega } & 1 \cr 0 & { - \omega } & { - \omega + 1} \cr } } \right]\end{aligned} where $$\omega = {{ - 1 + i\sqrt 3 } \over 2}$$, and I 3 be the identity matrix of order 3. If the determinant of the matrix (P $$-$$1 AP$$-$$I 3 ) 2 is $$\alpha$$$$\omega$$ 2 , then the value of $$\alpha$$ is equal to ______________.