Matrices practice · Medium

If $A,B,C$ are the angles of a triangle, then the determinant $\Delta = \left| {\begin{array}{cccccccccccccccccccc} {\sin 2A}&{\sin C}&{\sin B}\\ {\sin C}&{\sin 2B}&{\sin A}\\ {\sin B}&{\sin A}&{\sin 2C} \end{array}} \right|$ is equal to

Let $A=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right]$ and $B=I+\operatorname{adj}(A)+(\operatorname{adj} A)^2+\ldots+(\operatorname{adj} {A})^{10}$. Then, the sum of all the elements of the matrix $B$ is:

The matrix $A$ satisfying $A\left[\begin{array}{ll}1 & 5 \\ 0 & 1\end{array}\right]=\left[\begin{array}{cc}3 & -1 \\ 6 & 0\end{array}\right]$ is

If $f(x)=a+b x+c x^2$ and $\alpha, \beta, \gamma$ are roots of $x^3-1=0$, then $\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|$ is equal to

The function $f\left( x \right)$ given by $f\left( x \right) = \left| {\begin{array}{cccccccccccccccccccc} {x - 1\;x + 1\;2x + 1}\\ {x + 1\;x + 3\;2x + 3}\\ {2x + 1\;2x - 1\;4x + 1} \end{array}} \right|$ has

The value of $\left| {\begin{array}{cccccccccccccccccccc} {b + c}&a&a\\ b&{c + a}&b\\ c&c&{a + b} \end{array}} \right|,$ is

If $a_1, a_2, a_3, \ldots \ldots, a_n, \ldots$. are in G.P., then the value of the determinant
$\left|\begin{array}{ccc}\log a_n & \log a_{n+1} & \log a_{n+2} \\ \log a_{n+3} & \log a_{n+4} & \log a_{n+5} \\ \log a_{n+6} & \log a_{n+7} & \log a_{n+8}\end{array}\right|$, is

If $2 x-y=\left[\begin{array}{ccc}3 & -3 & 0 \\ 3 & 3 & 2\end{array}\right]$ and $2 y+x=\left[\begin{array}{ccc}4 & 1 & 5 \\ -1 & 4 & -4\end{array}\right]$, then

If ${a_1},{a_2}_. \ldots ..,{a_n}, \ldots ..,$ are in GP, then the determinant $\Delta = \left| {\begin{array}{cccccccccccccccccccc} {\log {a_n}}&{\log {a_{n + 1}}}&{\log {a_{n + 2}}}\\ {\log {a_{n + 3}}}&{\log {a_{n + 4}}}&{\log {a_{n + 5}}}\\ {\log {a_{n + 6}}}&{\log {a_{n + 7}}}&{\log {a_{n + 8}}} \end{array}} \right|$ is equal to

If adj $\mathrm{B}=\mathrm{A},|\mathrm{P}|=|\mathrm{Q}|=1$, then adj $\left(\mathrm{Q}^{-1} \mathrm{BP}^{-1}\right)$ is

If $\Delta_{1}=\left|\begin{array}{ccc}x & \sin \theta & \cos \theta \\ -\sin \theta & -x & 1 \\ \cos \theta & 1 & x\end{array}\right|$ and $\Delta_{2}=\left|\begin{array}{ccc}x & \sin 2 \theta & \cos 2 \theta \\ -\sin 2 \theta & -x & 1 \\ \cos 2 \theta & 1 & x\end{array}\right|, x \neq 0$; then for all $\theta \in\left(0, \frac{\pi}{2}\right)$

Consider the following relation $\mathrm{R}$ on the set of real square matrices of order 3 .
$R=\left\{(A, B) \mid A=P^{-1} B P\right.$ for some invertible matrix $\left.P\right\}$
Statement-1: $R$ is equivalence relation.

Statement-2 : For any two invertible $3 \times 3$ matrices $M$ and
$N,(M N)^{-1}=N^{-1} M^{-1} \text {. }$

If the system equations $(\lambda-1) x+(3 \lambda+1) y+2 \lambda z=0$, $(\lambda-1) x+(4 \lambda-2) y+(\lambda+3) z=0$, and $2 x+(3 \lambda+1) y+3(\lambda-1) z=0$ has non-trivial solutions for some values of $\lambda$, then the ratio $x: y: z$ when $\lambda$ has the smallest of these values, is

If A and B are non singular Matrices of same order then Adj. $(\mathrm{AB})$ is

In the interval $\left[ { - \frac{\pi }{4},\frac{\pi }{4}} \right],$ the number of real solutions of the equations
$\left| {\begin{array}{cccccccccccccccccccc} {\sin x}&{\cos x}&{\cos x}\\ {\cos x}&{\sin x}&{\cos x}\\ {\cos x}&{\cos x}&{\sin x} \end{array}} \right| = 0$ is

The value of $\left| {\begin{array}{cccccccccccccccccccc} {441}&{442}&{443}\\ {445}&{446}&{447}\\ {449}&{450}&{451} \end{array}} \right|$ is

If $A = \left[ {\begin{array}{cccccccccccccccccccc} \mathrm{\alpha }&0\\ 1&1 \end{array}} \right]$ and $B = \left[ {\begin{array}{cccccccccccccccccccc} 1&0\\ 5&1 \end{array}} \right]$, then value of α for which ${A^2} = B$ is

Determinant $\left| {\begin{array}{cccccccccccccccccccc} {1/a}&{{a^2}}&{bc}\\ {1/b}&{{b^2}}&{ca}\\ {1/c}&{{c^2}}&{ab} \end{array}} \right|$ is equal to

Let $A$ and $B$ be real matrices of the form $\left[\begin{array}{ll}\alpha & 0 \\ 0 & \beta\end{array}\right]$ and
$\left[\begin{array}{ll} 0 & \gamma \\ \delta & 0 \end{array}\right] \text {, respectively.}$

Statement 1: $A B-B A$ is always an invertible matrix.

Statement 2: $A B-B A$ is never an identity matrix

If $\Delta_1=\left|\begin{array}{ccc}1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2\end{array}\right|, \Delta_2=\left|\begin{array}{lll}1 & b c & a \\ 1 & c a & b \\ 1 & a b & c\end{array}\right|,$ then

If the system of linear equations
$x+a y+z=3$
$x+2 y+2 z=6$
$x+5 y+3 z=b$
has no solution, then

If $A=\left[\begin{array}{cc}\sqrt{2} & 1 \\ -1 & \sqrt{2}\end{array}\right], B=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right], C=A B A^T$ and $X=A^T C^2 A$, then det $X$ is equal to

Let $A = \left[ {\begin{array}{cccccccccccccccccccc} {{x^2}}&6&8 \\ 3&{{y^2}}&9 \\ 4&5&{{z^2}} \end{array}} \right],\,\,B = \left[ {\begin{array}{cccccccccccccccccccc} {2x}&3&5 \\ 2&{2y}&6 \\ 1&4&{2z - 3} \end{array}} \right]$ . If ${\text{trace}}\,A = {\text{trace}}\,B$, then $2x + y + x$ is equal to

The repeated factor of the determinant $\left| {\begin{array}{cccccccccccccccccccc} {y + z}&x&y\\ {z + x}&z&x\\ {x + y}&y&z \end{array}} \right|,$ is

If $A = \left[ {\begin{array}{cccccccccccccccccccc} \alpha &2\\ 2&\alpha \end{array}} \right]$ and $\left| {{A^3}} \right| = 125,$ then the value of $\alpha$ is

Which of the following is correct?

If $f(x)=\left|\begin{array}{lll}\cos (x+\alpha) & \cos (x+\beta) & \cos (x+\gamma) \\ \sin (x+\alpha) & \sin (x+\beta) & \sin (x+\gamma) \\ \sin (\beta-\gamma) & \sin (\gamma-\alpha) & \sin (\alpha-\beta)\end{array}\right|$, then $f(\theta ) - 2f(\phi ) + f(\psi )$ is equal to

If $f(\theta)=\left|\begin{array}{ccc}1 & \tan \theta & 1 \\ -\tan \theta & 1 & \tan \theta \\ -1 & -\tan \theta & 1\end{array}\right|$, then the set $\left\{f(\theta): 0 \leq \theta \leq \frac{\pi}{2}\right\}$ is

If $A, B, C$ are the angles of a non right angled triangle $A B C$, $\left|\begin{array}{ccc}\tan \mathrm{A} & 1 & 1 \\ 1 & \tan \mathrm{B} & 1 \\ 1 & 1 & \tan \mathrm{C}\end{array}\right|$ is equal to - Then the value of

If $A = \left[ {\begin{array}{cccccccccccccccccccc} 1&{ - 1}&1\\ 0&2&{ - 3}\\ 2&1&0 \end{array}} \right]$ and $\;B = \left( {\mathrm{adj}\;A} \right),\;$ and $C = 5A,\;$ then $\;\frac{{\left| {\mathrm{adj}\;B} \right|}}{{\left| C \right|}}\;$ is equal to