Matrices practice · Medium

In $\Delta \;ABC,\mathrm{if}\left| {\begin{array}{cccccccccccccccccccc} 1&a&b\\ 1&c&a\\ 1&b&c \end{array}} \right| = 0$, then

${\sin ^2}A + {\sin ^2}B + {\sin ^2}C$ is equal to

Let $p$ be an odd prime number and $T_p$ be the following set of $2 \times 2$ matrices $T_p=\left\{A=\left[\begin{array}{ll}a & b \\ c & a\end{array}\right]: a, b, c \in\{0,1,2, \ldots,(p-1)\}\right\}$
The number of $A$ in $T_p$ such that $A$ is either symmetric or skew-symmetric or both, and det $(A)$ divisible by $p$ is

$\left|\begin{array}{lll}1 & \mathrm{a} & \mathrm{a}^{2} \\ 1 & \mathrm{b} & \mathrm{b}^{2} \\ 1 & \mathrm{c} & \mathrm{c}^{2}\end{array}\right|=$

Consider the following statements

I. $|{A}|$ is also called modulus of square matrix $A.$

II. Every matrix has determinant.

If $\alpha ,\beta$ are the roots of equation $a{x^2} + bx + c = 0,$ then the value of the determinant
$\left| {\begin{array}{cccccccccccccccccccc} 1&{\cos \;\left( {\beta - \alpha } \right)}&{\cos \alpha }\\ {\cos \;\left( {\alpha - \beta } \right)}&1&{\cos \beta }\\ {\cos \alpha }&{\cos \beta }&1 \end{array}} \right|,$ is

Let $a,b,c$ be positive real numbers. The following system of equations in $x,y$ and $z$ $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} - \frac{{{z^2}}}{{{c^2}}} = 1,\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1, - \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1$ has

The value of $f\left(\frac{\pi}{6}\right)$ where $f(q)=\left|\begin{array}{ccc}\cos ^2 \theta & \cos \theta \sin \theta & -\sin \theta \\ \cos \theta \sin \theta & \sin ^2 \theta & \cos \theta \\ \sin \theta & -\cos \theta & 0\end{array}\right|$ is

The value of the determinant $\left|\begin{array}{ccc}\cos \alpha & -\sin \alpha & 1 \\ \sin \alpha & \cos \alpha & 1 \\ \cos (\alpha+\beta) & -\sin (\alpha+\beta) & 1\end{array}\right|$ is

Consider the matrix $f(x)=\left[\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right]$
Given below are two statements:
Statement I: $f(-x)$ is the inverse of the matrix $f(x).$
Statement II : $f(x) f(y)=f(x+y).$
In the light of the above statements, choose the correct answer from the options given below.

The solutions of the equation $\left| {\begin{array}{cccccccccccccccccccc} x&2&{ - 1}\\ 2&5&x\\ { - 1}&2&x \end{array}} \right| = 0,$ are

If $A=\left[\begin{array}{ccc}-1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 & -1\end{array}\right]$, then the value of $\operatorname{Det}(A B+2I)$ where $B=\operatorname{adj} A$ and $I$ is identify matrix of order 3

If $\left|\begin{array}{lll}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right| \neq 0$, then the number of solutions of the system of equations $a_1 x+b_1 y+c_1 z=0, a_2 x+b_2 y+c_2 z=0$ and $a_3 x+b_3 y+c_3 z=0$ is

If maximum and minimum values of the determinant $\left|\begin{array}{lll}1+\operatorname{Sin}^2 x & \operatorname{Cos}^2 x & \operatorname{Sin} 2 x \\ \operatorname{Sin}^2 x & 1+\operatorname{Cos}^2 x & \operatorname{Sin} 2 x \\ \operatorname{Sin}^2 x & \operatorname{Cos}^2 x & 1+\operatorname{Sin} 2 x\end{array}\right|$ are $\alpha$ and $\beta$ respectively, then which of the statements are FALSE ?

The value of $\left| {\begin{array}{cccccccccccccccccccc} {x + y}&{y + z}&{z + x}\\ x&y&z\\ {x - y}&{y - z}&{z - x} \end{array}} \right|$ is equal to

If $\mathrm{A}$ is $3 \times 3$ matrix and if $|\mathrm{A}|=3$, then $|\operatorname{adj} \mathrm{A}|=$

Let $a, b, c$ be such that $b(a+c) \neq 0$ if
$\left|\begin{array}{ccc} a & a+1 & a-1 \\ -b & b+1 & b-1 \\ c & c-1 & c+1 \end{array}\right|+\left|\begin{array}{ccc} a+1 & b+1 & c-1 \\ a-1 & b-1 & c+1 \\ (-1)^{n+2} a & (-1)^{n+1} b & (-1)^n c \end{array}\right|=0 \text {, }$
then the value of $n$ is :

If $a>0$ and discriminant of $a x^2+2 b x+c$ is negative, then
$\Delta=\left|\begin{array}{ccc}a & b & a x+b \\ b & c & b x+c \\ a x+b & b x+c & 0\end{array}\right|,$ is

If ${\Delta _a} = \left| {\begin{array}{cccccccccccccccccccc} {a - 1}&n&6\\ {{{\left( {a - 1} \right)}^2}}&{2{n^2}}&{4n - 2}\\ {{{\left( {a - 1} \right)}^3}}&{3{n^3}}&{3{n^2} - 3n} \end{array}} \right|,$ then $\sum\nolimits_{a = 1}^n {{\Delta _a}}$ is equal to

$\left| {\begin{array}{cccccccccccccccccccc} {a - b + c}&{ - a - b + c}&1\\ {a + b + 2c}&{ - a + b + 2c}&2\\ {3c}&{3c}&3 \end{array}} \right|$ is

Matrix $A$ is such that ${A^2} = 2A - I$ where $I$ is the identity matrix, then for $n \ge 2,\;{A^n}$ is equal to

If $A = \left[ {\begin{array}{cccccccccccccccccccc} 1&2\\ 3&5 \end{array}} \right],$ then the value of the determinant $\left| {{A^{2009}} - 5{A^{2008}}} \right|$

Consider the system of linear equations;

$x_1+2 x_2+x_3=3$

$2 x_1+3 x_2+x_3=3$

$3 x_1+5 x_2+2 x_3=1$

The system has

None of these $\left| {\begin{array}{cccccccccccccccccccc} {x - 1}&1&1\\ 1&{x - 1}&1\\ 1&1&{x - 1} \end{array}} \right| = 0,$ are

If $A = \left[ {\begin{array}{cccccccccccccccccccc} 1&2\\ 2&1 \end{array}} \right]$ and $f\left( x \right) = \frac{{1 + x}}{{1 - x}}$, then $f\left( A \right)$ is

Determinant $\left| {\begin{array}{cccccccccccccccccccc} 1&x&y\\ 2&{\sin x + 2x}&{\sin y + 3y}\\ 3&{\cos x + 3x}&{\cos y + 3y} \end{array}} \right|$ is equal to

The factors of $\left| {\begin{array}{cccccccccccccccccccc} x&a&b\\ a&x&b\\ a&b&x \end{array}} \right|$ are

Let the matrix $A=\left|\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{array}\right|$ and the matrix $B_0$ $=A^{49}+2 A^{98}$. If $B_n=\operatorname{Adj}\left(B_{n-1}\right)$ for all $n \geq 1$, then $\operatorname{det}\left(B_4\right)$ is equal to :

If the system of equations
$\begin{array}{l} x + a\mathrm{ }y + a\mathrm{ }z = 0\\ b\mathrm{ }x + y + b\mathrm{ }z = 0\\ c\mathrm{ }x + c\mathrm{ }y + z = 0 \end{array}$
Where $a,b$ and $c$ are non-zero non-unity, has a non-trivial solution, then the value of $\frac{a}{{1 - a}} + \frac{b}{{1 - b}} + \frac{c}{{1 - c}}$ is

If the product of $n$ matrices $\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}1 & 3 \\ 0 & 1\end{array}\right] \ldots\left[\begin{array}{ll}1 & n \\ 0 & 1\end{array}\right]$ is equal to the matrix $\left[\begin{array}{cc}1 & 378 \\ 0 & 1\end{array}\right]$, then the value of $n$ is equal to

Let ${A}$ and ${B}$ be two matrices such that $A=\left[a_{i j}\right]_{2 \times 3}=\left[\begin{array}{ccc}x & 1 & 2 \\ 3 x & -1 & \frac{1}{4 x^2+1}\end{array}\right]$ and $B=\left[b_{i j}\right]_{3 \times 2}=\left[\begin{array}{cc}\frac{1}{x^2} & \frac{1}{x} \\ 2 x & 2 \\ 3 & x\end{array}\right]$ where $x > 0$. If matrix $C$ is defined as $C=\left[c_{i j}\right]_{2 \times 2}$, where $c_{i j}=\sum\limits_{{r=1}}^3 a_{i r} b_{r j}, \forall 1 \leq i, j \leq 2$, then the minimum value of $\Delta(x)=\sum\limits_{{1 \leq i \leq j \leq 2}} c_{i j}$ is