Matrices practice · Mixed difficulty

If $a_1\, b_1\, c_1, a_2\, b_2\, c_2$ and $a_3\, b_3\, c_3$ are three digit even natural numbers and
$\Delta=\left|\begin{array}{lll}c_1 & a_1 & b_1 \\ c_2 & a_2 & b_2 \\ c_3 & a_3 & b_3\end{array}\right|$, then $\Delta$ is

Let $P = \left[ {\begin{array}{cccccccccccccccccccc} 1&0&0\\ 3&1&0\\ 9&3&1 \end{array}} \right]\mathrm{ }\,\,\mathrm{and}\,\,\mathrm{ }Q = \left[ {{q_{ij}}} \right]\mathrm{ }\,\,\mathrm{be}\,\,\mathrm{ two }\,\,3 \times 3$ matrices such that $Q - {P^5} = {I_3}.$ Then, $\frac{{{q_{21}} + {q_{31}}}}{{{q_{32}}}}$ is equal to

If $\Delta=\left|\begin{array}{ccc}\cos \frac{\theta}{2} & 1 & 1 \\ 1 & \cos \frac{\theta}{2} & -\cos \frac{\theta}{2} \\ -\cos \frac{\theta}{2} & 1 & -1\end{array}\right|$ lies in the interval

If for a matrix $A,{A^2} + I = O,$ where $I$ is the identity matrix, then $A$ equals

The matrices $P = \left[ {\begin{array}{cccccccccccccccccccc} {{u_1}}&{{v_1}}&{{w_1}}\\ {{u_2}}&{{v_2}}&{{w_2}}\\ {{u_3}}&{{v_3}}&{{w_3}} \end{array}} \right];Q = \frac{1}{9}\left[ {\begin{array}{cccccccccccccccccccc} 2&2&1\\ {13}&{ - 5}&m\\ { - 8}&1&5 \end{array}} \right]$ are such that $PQ = I$, an identity matrix. Solving the equation $\left[ {\begin{array}{cccccccccccccccccccc} {{u_1}}&{{v_1}}&{{w_1}}\\ {{u_2}}&{{v_2}}&{{w_2}}\\ {{u_3}}&{{v_3}}&{{w_3}} \end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc} x\\ y\\ z \end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc} 1\\ 1\\ 5 \end{array}} \right]$, the value of $y$ comes out to be $- 3$. Then, the value of $m$ 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

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

Two fair dice are thrown. The numbers on them are taken as $\lambda$ and $\mu$, and a system of linear equations $x + y + z = 5,\,\,x + 2y + 3z = \mu$ and $x + 3y + \lambda z = 1$ is constructed. If $p$ is the probability that the system has a unique solution and $q$ is the probability that the system has no solution, then

Let $A, B, C$ be three square matrices of the same order, such that whenever $A B=A C$ then $B=C$, if $A$ is

Which one of the following statement is true?

If the system of linear equations.
$\begin{gathered} 8x + y + 4z = - 2 \quad \\ x + y + z = 0 \quad \\ \lambda x - 3y = \mu \quad \\ \end{gathered}$
has infinitely many solutions, then the distance of the point $\left(\lambda, \mu,-\frac{1}{2}\right)$ from the plane $8 x+y+4 z+2=0$ is

Consider the following system of equations
$x + 2y - 3z = a$
$2x + 6y - 11z = b$
$x - 2y + 7z = c$
Where, $a,\,b$ and $c$ are real constants. Then, the system of equations

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

The matrix $A=\left(\begin{array}{cc}p & -q \\ q & p\end{array}\right)$ is orthogonal if and only if

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

If $x,y,z$ are in A.P., then the value of the $\det \left( A \right)$ is, where $A = \left[ {\begin{array}{cccccccccccccccccccc} 4&5&6&x\\ 5&6&7&y\\ 6&7&8&z\\ x&y&z&0 \end{array}} \right]$

Which of the given values of $\mathrm{x}$ and $y$ make the following pairs of matrices equal:
$\left[\begin{array}{cc} 3 x+7 & 5 \\ y+1 & 2-3 x \end{array}\right] \cdot\left[\begin{array}{cc} 0 & y-2 \\ 8 & 4 \end{array}\right]$

In how many ways, the determinant of order 3 can be expanded?

If $\left| {\begin{array}{cccccccccccccccccccc} {6i}&{ - 3i}&1\\ 4&{3i}&{ - 1}\\ {40}&3&i \end{array}} \right| = x + iy$, then

Let $\alpha$ and $\beta$ be the roots of the equation ${x^2} + x + 1 = 0.$ Then, for $y \ne 0$ in $R,\,\,\left| {\begin{array}{cccccccccccccccccccc} {y + 1}&\alpha &\beta \\ \alpha &{y + \beta }&1\\ \beta &1&{y + \alpha } \end{array}} \right|$ is equal to

If $A=\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]$ and $A^2=\left[\begin{array}{ll}\alpha & \beta \\ \beta & \alpha\end{array}\right]$, then

If $p+q+r=0$ and $\left|\begin{array}{lll}p a & q b & r c \\ q c & r a & p b \\ r b & p c & q a\end{array}\right|=K\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|$ then the value of $K$ is

Let $A$ be a $3 \times 3$ matrix such that $A^2-5 A+7 I=O$.

Statement $-I: A^{-1}=\frac{1}{7}(5 I-A)$.

Statement $-I I$ : The polynomial $A^3-2 A^2-3 A+I$ can be reduced to $5(A-4 I)$. Then

If $\mathrm{A}, \mathrm{B}$ are two square matrices such that $\mathrm{AB}=\mathrm{A}$ and $\mathrm{BA}=\mathrm{B}$, then

If each of third order determinant of value $\Lambda$ is multiplied by 4 , then value of the new determinant is:

The adjoint matrix of $\left[ {\begin{array}{cccccccccccccccccccc} 3&{ - 3}&4\\ 2&{ - 3}&4\\ 0&{ - 1}&1 \end{array}} \right]$ is

If $- 9$ is a root of the equation $\left| {\begin{array}{cccccccccccccccccccc} x&3&7\\ 2&x&2\\ 7&6&x \end{array}} \right| = 0$, then the other two roots are

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

If $X, Y$ and $Z$ are positive numbers such that $Y$ and $Z$ have respectively 1 and 0 at their unit's place and
$\Delta = \left| {\begin{array}{llllllllllllllllllll} X&4&1 \\ Y&0&1 \\ Z&1&0 \end{array}} \right|$
If $(\Delta+1)$ is divisible by 10 , then $X$ has at its unit's place