Matrices practice · Mixed difficulty

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

From the matrix equation $AB = AC$ we can conclude $B = C$ provided that

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?

If $A=\left[\begin{array}{rrr}2 & 1 & -1 \\ 3 & 5 & 2 \\ 1 & 6 & 1\end{array}\right]$, then $\operatorname{tr}(\operatorname{Aadj}(\operatorname{adj} \mathrm{A})$) is equal to (where $\operatorname{tr}(\mathrm{P})$ denotes the trace of the matrix P i.e the sum of all the diagonal elements of the matrix P and adj (P) denotes the adjoint of matrix P)

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

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

Let the numbers 2, b, c be in an AP and
$A = \left[ {\begin{array}{cccccccccccccccccccc} 1&1&1\\ 2&b&c\\ 4&{{b^2}}&{{c^2}} \end{array}} \right].\mathrm{ If}\det (A) \in [2,16],\mathrm{then}\,\mathrm{ }c$ lies in the interval

A root of the equation $\left| {\begin{array}{cccccccccccccccccccc} 0&{x - a}&{x - b}\\ {x + a}&0&{x - c}\\ {x + b}&{x + c}&0 \end{array}} \right| = 0,$ is

If $A = \left[ {\begin{array}{cccccccccccccccccccc} 3&2\\ 0&1 \end{array}} \right],\;$ then ${\left( {{A^{ - 1}}} \right)^3}$ is equal to

Let $\alpha$ be a root of the equation ${x^2} + x + 1 = 0$ and the matrix $A = \frac{1}{{\sqrt 3 }}\left[ {\begin{array}{cccccccccccccccccccc} 1&1&1\\ 1&\alpha &{{\alpha ^2}}\\ 1&{{\alpha ^2}}&{{\alpha ^4}} \end{array}} \right]$, then the matrix ${A^{31}}$ is equal to

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

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

If $a, b$ and $c$ are non-zero real numbers and if the equations $(a-1) x=y+z,(b-1) y=z+x$, $(c-1) z=x+y$ has a non-trivial solution, then $a b+b c+c a$ equals to

Let $A = \left[ {\begin{array}{cccccccccccccccccccc} 1&2&{ - 3}\\ { - 2}&0&3\\ 3&{ - 3}&1 \end{array}} \right]$ be a matrix, then (determinant of A)$\times$ (adjoint of inverse A) is equal to

If $\omega$ be a complex cube root of unity, then $\left| {\begin{array}{cccccccccccccccccccc} 1&\omega &{ - {\omega ^2}/2}\\ 1&1&1\\ 1&{ - 1}&0 \end{array}} \right|$ is equal to

Let $\left[ x \right]$ represent the greatest integer less than or equal to $x$, then the value of the determinant $\left| {\begin{array}{cccccccccccccccccccc} {\left[ e \right]}&{\left[ \pi \right]}&{\left[ {{\pi ^2} - 6} \right]}\\ {\left[ \pi \right]}&{\left[ {{\pi ^2} - 6} \right]}&{\left[ e \right]}\\ {\left[ {{\pi ^2} - 6} \right]}&{\left[ e \right]}&{\left[ \pi \right]} \end{array}} \right|$ is

Suppose $p, q, r \neq 0$ and system of equations

has a non-trivial solution, then value of $\frac{a}{p}+\frac{b}{q}+\frac{c}{r}$ is

If $A$ is a matrix such that ${A^2} = A + 1,\;$ where $I$ is the unit matrix, then ${A^5}\;$ is equal to

Let $P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right], A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ and $Q=P A P^{T}$. If $P^{T}$ $Q^{2007} P=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$, then $2 a+b-3 c-4 d$ equal to

A root of the equation $\left| {\begin{array}{cccccccccccccccccccc} {3 - x}&{ - 6}&3\\ { - 6}&{3 - x}&3\\ 3&3&{ - 6 - x} \end{array}} \right| = 0,$ is

If $S$ is the set of distinct values of $'b'$ for which the following system of linear equations
$x+y+z=1$

$x+a y+z=1$

$a x+b y+z=0$
has no solution, then $S$ is

If $a_1, a_2, a_3, \ldots, a_n, \ldots$ are in G.P., then the value of the determinant $\left|\begin{array}{ccc} \log a_n & \log a_{n+1} & \log a_{n+2} \\ \log a_{n+1} & \log a_{n+2} & \log a_{n+3} \\ \log a_{n+2} & \log a_{n+3} & \log a_{n+4} \end{array}\right|$ is

If $A=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1\end{array}\right]$, then the trace of the matrix $A$ 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