Matrices practice · Easy

If $\mathrm{C}=2 \cos \theta,$ then the value of the determinant $\Delta=\left|\begin{array}{lll}C & 1 & 0 \\ 1 & C & 1 \\ 6 & 1 & C\end{array}\right|,$ is

If $A$ and $B$ are two square matrix such that $A^{2} B=B A$ and if $(A B)^{10}=A^{k} B^{10}$, then $k$ is

Study the following statement and select the correct option:
Statement-I: If $a, b, c$ are in A.P the system of equations $3 x+4 y+5 z=a, 4 x+5 y+6 z=b, 5 x+6 y+7 z=c$ is consistent

Statement-II: If $|A| \neq 0$, the system of equations $A X=B$ is consistent

Let $A=\left(\begin{array}{cc}2 & -1 \\ 0 & 2\end{array}\right)$. If $B=I-{ }^5 C_1(\operatorname{adj} A)+{ }^5 C_2(\operatorname{adj} A)^2-$ .....$-{ }^5 C_5(\operatorname{adj} A)^5$, then the sum of all elements of the matrix $B$ is

Let $A_t=\left(\begin{array}{ccc}1 & 3 & 2 \\ 2 & 5 & t \\ 4 & 7-t & -6\end{array}\right)$, then the value(s) of $t$ for which inverse of $A_t$ does not exist.

The matrix $\left[\begin{array}{lll}1 & 0 & 1 \\ 2 & 1 & 0 \\ 3 & 1 & 1\end{array}\right]$ is

If $a_{i j}=\frac{(i-2 j)^2}{3}$, then $\mathrm{A}=\left[a_{i j}\right]_{2 \times 2}=$

If $\alpha$ is a complex root of the equation $x^2+x+1=0$ and $A=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1 & 1 & 1 \\ 1 & \alpha & \alpha^2 \\ 1 & \alpha^2 & \alpha\end{array}\right]$, then $A^{31}$ is equal to

If the system of equations $2 x-y+z=0, x-2 y+z=0$, $t x-y+2 z=0$ has infinitely many solutions and $f(x)$ be a continuous function, such that $f(5+x)+f(x)=2$, then $\int_0^{-2 t} f(x) d x$ is equal to

If $a_0, a_1, a_2, a_3, a_4$ are in A.P. with the common difference $d$,
the value of $\left|\begin{array}{lll}a_1 a_2 & a_1 & a_0 \\ a_2 a_3 & a_2 & a_1 \\ a_3 a_4 & a_3 & a_2\end{array}\right|$ is

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

If $X = \left[ {\begin{array}{cccccccccccccccccccc} 3&{ - 4}\\ 1&{ - 1} \end{array}} \right]$, then the value of ${X^n}$ is

Let $A$ be a skew-symmetric matrix of order $n$ then

If $A=\left[\begin{array}{ll}3 & 2 \\ 0 & 1\end{array}\right]$, then $A^{-3}$ is

If $A = \left[ {\begin{array}{cccccccccccccccccccc} {1\; - 5\;\;\;\;7}\\ {0\;\;\;\;\;7\;\;\;\;9}\\ {11\;\;\;\;\;8\;\;\;9} \end{array}} \right]$, then trace of matrix $A$ is

If $A=\left[\begin{array}{lll}n & 0 & 0 \\ 0 & n & 0 \\ 0 & 0 & n\end{array}\right]$ and $B=\left[\begin{array}{lll}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{array}\right]$, then $A B$ equal to

If $A=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$, then $A^{100}=$

Let $X = \left[ {\begin{array}{cccccccccccccccccccc} {{x_1}}\\ {{x_2}}\\ {{x_3}} \end{array}} \right],\;A = \left[ {\begin{array}{cccccccccccccccccccc} 1&{ - 1}&2\\ 2&0&1\\ 3&2&1 \end{array}} \right]$ and $B = \left[ {\begin{array}{cccccccccccccccccccc} 3\\ 1\\ 4 \end{array}} \right]$. If $AX = B$, then $X$ is equal to

If $A$ and $B$ are $2 \times 2$ matrices, then which of the following is true?

Let $A=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & -2 & 4\end{array}\right]$

If
$A^{-1}=\frac{1}{6}\left(A^2+x A+y I\right)$ where $x, y$ are scalars and $I$ is $3 \times 3$ unit matrix, then
$x, y$ are, respectively

The inverse of a skew symmetric matrix of odd order is

Consider the following statements

I. For multiplication of two matrices $A$ and $B$, the number of columns in $A$ should be less than the number of rows in ${B}$.

II. For getting the elements of the product matrix, we take rows of $A$ and column of $B$, multiply them elementwise and take the sum.

Choose the correct option.

If $\mathrm{A}^3=0$ then $\mathrm{I}+\mathrm{A}+\mathrm{A}^2$ equals:-

If $A$ and $B$ are square matrices such that $A^{2006}=0$ and $A B=A+B,$ then $\operatorname{det}(B)$ equals to

If ${A}=\left[\begin{array}{ll}2 & 2 \\ 2 & 5\end{array}\right]$ and ${B}=\left[\begin{array}{cc}2 & 1 \\ -1 & 2\end{array}\right]$, then ${B}^{-1} {AB}=$

Choose the correct answer in the following questions : If $A, B$ are symmetric matrices of same order then $\mathrm{AB}-\mathrm{BA}$ is a

If $I=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right), J=\left(\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right)$, and $B=\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right)$, then $B$ equals

The matrix $A = \left[ {\begin{array}{cccccccccccccccccccc} {\frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}}\\ { - \frac{1}{{\sqrt 2 }}}&{ - \frac{1}{{\sqrt 2 }}} \end{array}} \right]$ is

If $\left[\begin{array}{ll}2 & 1 \\ 7 & 4\end{array}\right] A\left[\begin{array}{rr}-3 & 2 \\ 5 & -3\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$, then matrix $A$ equals

If ${P}=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right], {A}=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ and ${Q}={PA} {P}^{\top}$, then ${P}^{\top}\left({Q}^{2005}\right) {P}$ is equal to