Matrices practice · Easy

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 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

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

Let $A=\left[\begin{array}{ccc}2 & 1 & 0 \\ 1 & 2 & -1 \\ 0 & -1 & 2\end{array}\right].$ If $|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} 2 A))|=(16)^n,$ then $n$ is equal to

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.

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

If $A$ is a $3 \times 3$ non-singular matrix, then $\operatorname{adj}(\operatorname{adj} A)$ is equal to

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

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

$\left[\begin{array}{lll}7 & 1 & 2 \\ 9 & 2 & 1\end{array}\right]\left[\begin{array}{l}3 \\ 4 \\ 5\end{array}\right]+2\left[\begin{array}{l}4 \\ 2\end{array}\right]$ is equal to

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 $\left| {\begin{array}{cccccccccccccccccccc} {y + z}&x&y\\ {z + x}&z&x\\ {x + y}&y&z \end{array}} \right| = k\left( {x + y + z} \right){\left( {x - z} \right)^2},$ then $k$ is equal to

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

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

Let $A\left( {\begin{array}{cccccccccccccccccccc} 0&{2q}&r\\ p&q&{ - r}\\ p&{ - q}&r \end{array}} \right)$. If $A{A^T} = {I_3},$ then $\left| p \right|$ is

If $A = \left[ {\begin{array}{cccccccccccccccccccc} {0\;\;1}\\ {0\;\;0} \end{array}} \right],I$ is the unit matrix of order 2 and $a,b$ are arbitrary constants, then ${\left( {aI + bA} \right)^2}$ is equal to

If $\left[ {\begin{array}{cccccccccccccccccccc} 1&{ - \tan \theta }\\ {\tan \theta }&1 \end{array}} \right]{\left[ {\begin{array}{cccccccccccccccccccc} 1&{\tan \theta }\\ { - \tan \theta }&1 \end{array}} \right]^{ - 1}} = \left[ {\begin{array}{cccccccccccccccccccc} a&{ - b}\\ b&a \end{array}} \right],$ then

Which one of the following is true always for any two non-singular matrices $A$ and $B$ of same order?

If $\mathrm{A}=\left[\begin{array}{ll}a & b\end{array}\right], \mathrm{B}=\left[\begin{array}{ll}-b & -a\end{array}\right]$ and $\mathrm{C}=\left[\begin{array}{c}a \\ -a\end{array}\right]$, then the correct statement is

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

$\cos \theta \left[ {\begin{array}{cccccccccccccccccccc} {\cos \theta \;\;\;\;\sin \theta }\\ { - \sin \theta \;\;\;\;\;\cos \theta } \end{array}} \right] + \sin \theta \left[ {\begin{array}{cccccccccccccccccccc} {\sin \theta - \cos \theta }\\ {\cos \theta \;\;\;\;\sin \theta } \end{array}} \right]$ is equal to