Matrices practice · Mixed difficulty

Let m and M be respectively the minimum and maximum values of $\left| {\begin{array}{cccccccccccccccccccc} {{{\cos }^2}x}&{1 + {{\sin }^2}x}&{\sin 2x}\\ {1 + {{\cos }^2}x}&{{{\sin }^2}x}&{\sin 2x}\\ {{{\cos }^2}x}&{{{\sin }^2}x}&{1 + \sin 2x} \end{array}} \right|.$ Then, the ordered pair $(m,M)$ is equal to

Consider the following statements

I. To every rectangular matrix $A=\left[a_{i j}\right]$ of order $n$, we can associate a number (real or complex) called determinant of $A.$

II. Determinant is a function which associates each square matrix with a unique number (real or complex).

If $A = \left[ {\begin{array}{cccccccccccccccccccc} {6\;\;8\;\;5}\\ {4\;\;2\;\;3}\\ {9\;\;7\;\;1} \end{array}} \right]$ is the sum of a symmetric matrix $B$ skew-symmetric matrix $C$, then $B$ is

If $A$ is an orthogonal matrix, then $A^{-1}$, equals

If $U = \left[ {2 - 3\;\;\;4} \right],X = \left[ {0\;\;2\;\;3} \right],V = \left[ {\begin{array}{cccccccccccccccccccc} 3\\ 2\\ 1 \end{array}} \right]\mathrm{and}\;Y = \left[ {\begin{array}{cccccccccccccccccccc} 2\\ 2\\ 4 \end{array}} \right],\mathrm{then}\;UV + XY$

Assertion: $\Delta=\left|\begin{array}{ccc}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ k a_1 & \mathrm{ka}_2 & \mathrm{ka}_3\end{array}\right|=0$

Reason: If corresponding elements of any two rows of a determinant are proportional, then its value is zero.

If $A=\left(\begin{array}{lll}3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1\end{array}\right)$ then value of trace of the matrix $\operatorname{adj}(\operatorname{adj} A)$ is

If the system of linear equations
$2x + 3y - z = - 2$
$x + y + z = 4$
$x - y + \left| \lambda \right|z = 4\lambda - 4$
where $\lambda \in R,$ has no solution, then

The square matrix $A=\left[a_{i j}\right]$ given by $a_{i j}=(i-j)^3$, is a

If $2A + 3B = \left[ {\begin{array}{cccccccccccccccccccc} {2\; - 1\;\;4}\\ {3\;\;\;\;\;\;2\;\;5} \end{array}} \right]\mathrm{and\;}A + 2B = \left[ {\begin{array}{cccccccccccccccccccc} {5\;\;0\;\;3}\\ {1\;\;6\;\;2} \end{array}} \right],\;\mathrm{then}\;\;B\;\mathrm{is}$

If $P$ is an orthogonal matrix and $Q=P A P^T$ and $B=P^T Q^{1000} P,$ then $B^{-1}$ is, where $A$ is involutory matrix

If $A$ is an orthogonal matrix, then $|A|$ is

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

The values of $\lambda$ and $\mu$ for which the system of linear equations

$\begin{aligned} &x+y+z=2 \\ &x+2 y+3 z=5 \\ &x+3 y+\lambda z=\mu\end{aligned}$

has infinitely many solutions are respectively

For any square matrix $A$ with real number entries, consider the following statements.

Assertion : $A$ + ${A}^{\prime}$ is a symmetric matrix.

Reason: ${A-A}^{\prime}$ is a skew-symmetric matrix.

Consider the following statements

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

II. Every matrix has determinant.

Let $A$ and $B$ be $3 \times 3$ real matrices, such that $A$ is symmetric matrix and $B$ is skew-symmetric matrix. Then, the system of linear equations $({A^2}{B^2} - {B^2}{A^2})X = 0$, where $X$ is a $3 \times 1$ column matrix of unknown variables and $O$ is a $3 \times 1$ null matrix, has

Consider the system of equations in $x,\;y,\;z$ as
$\begin{array}{l} x\sin 3\theta - y + z = 0\\ x\cos 2\theta + 4y + 3z = 0 \end{array}$
and $\;2x + 7y + 7z = 0$
If this system has a non-trivial solution, then for integer $n$, values of $\theta$ are given by

Inverse of the matrix $\left[ {\begin{array}{cccccccccccccccccccc} {\cos 2\mathrm{\theta }\;--\sin 2\mathrm{\theta }}\\ {\sin 2\mathrm{\theta }\;\cos 2\mathrm{\theta }} \end{array}} \right]$ 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

If the matrix ${M_r}$ is given by ${M_r} = \left[ {\begin{array}{cccccccccccccccccccc} r&{r - 1}\\ {r - 1}&r \end{array}} \right]r = 1,2,3...,$ then the value of det $\left( {{M_1}} \right) + \det \left( {{M_2}} \right) + ... + \det ({M_{2008}})$ is

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

If $O\left( A \right)\; = 2 \times 3,\;O\left( B \right)\; = 3 \times 2,\;$ and $\;O\left( c \right) = 3 \times 3,\;$ Which one of the following is not defined?

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 ${A^2} - A + I = 0,\;$ then the inverse of $A$ is

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 ?

If $A$ is a $2 \times 2$ unitary matrix, then $|A|$ is equal to

For all values of $\theta \in(0, \pi / 2),$ the determinant of the matrix $\left[\begin{array}{ccc}-2 & \tan \theta+\sec ^2 \theta & 3 \\ -\sin \theta & \cos \theta & \sin \theta \\ -3 & -4 & 3\end{array}\right]$ always lies in the interval