Matrices practice · Medium

If $a^2+b^2+c^2=-2$ and
$\mathrm{f}(\mathrm{x})=\left|\begin{array}{ccc} 1+a^2 x & \left(1+b^2\right) x & \left(1+c^2\right) x \\ \left(1+a^2\right) x & 1+b^2 x & \left(1+c^2\right) x \\ \left(1+a^2\right) x & \left(1+b^2\right) x & 1+c^2 x \end{array}\right|$
then $f(x)$ is a polynomial of degree

If $0 \leq[{x}] < 2 ;-1 \leq[{y}] < 1$ and $1 \leq[{z}] < 3$. where $[*]$ denotes the greatest integer function,

The value of $x$ obtained from the equation $\left| {\begin{array}{cccccccccccccccccccc} {x + \alpha }&\beta &\gamma \\ \gamma &{x + \beta }&\alpha \\ \alpha &\beta &{x + \gamma } \end{array}} \right| = 0$ will be

If $A$ is square matrix of order $n, a=$ maximum number of distinct entries. If $A$ is triangular matrix, $b=$ maximum number of distinct entries. If $A$ is a diagonal matrix. $c=$ minimum number of zeros. If $A$ is a triangular matrix if $a+5=c+2 b$

If $A = \left[ {\begin{array}{cccccccccccccccccccc} 3&4\\ 5&7 \end{array}} \right]$ then $A.\left( {\mathrm{adj}\;A} \right)$ is equal to

If $a, b, c$ are positive integers such that $a > b > c$ and $\left|\begin{array}{ccc} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{array}\right|=-2$ then $3 a+7 b-10 c$ equals

Let $A\left[\begin{array}{cc}1 & 2 \\ -5 & 1\end{array}\right]$ and $A^{-1}=x A+y I$, then the value of $(2 x+3 y)$ is______

If $\left[ {\begin{array}{cccccccccccccccccccc} 1&2&x\\ 4&{ - 1}&7\\ 2&4&{ - 6} \end{array}} \right]$ is a singular matrix, then $x$ is equal to

The value of the determinant $\left| {\begin{array}{cccccccccccccccccccc} x&a&{b + c}\\ x&b&{c + a}\\ x&c&{a + b} \end{array}} \right| = 0$, if

A factor of $\Delta \left( x \right) = \left| {\begin{array}{cccccccccccccccccccc} {{x^3} + 1}&{2{x^4} + 3{x^2}}&{3{x^5} + 4x}\\ 2&5&7\\ 3&{14}&{19} \end{array}} \right|$ is

If $A=\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r\end{array}\right]$ then $A^3-r A^2-q A=$

If ${D_r} = \left| {\begin{array}{cccccccccccccccccccc} {{2^{r - 1}}}&{{3^{r - 1}}}&{{4^{r - 1}}}\\ x&y&z\\ {{2^n} - 1}&{\left( {{3^n} - 1} \right)/2}&{\left( {{4^n} - 1} \right)/3} \end{array}} \right|$, then the value of $\sum\nolimits_{r = 1}^n {{D_r}}$ is equal to

One root of the equation $\left| {\begin{array}{cccccccccccccccccccc} x&{ - 6}&{ - 1}\\ 2&{ - 3x}&{x - 3}\\ { - 3}&{2x}&{x + 2} \end{array}} \right| = 0$ is

${\left[ {\begin{array}{cccccccccccccccccccc} {0\;\;a}\\ {b\;\;O} \end{array}} \right]^4} = I$, then

The number of integers $x$ satisfying
$-3 x^4+\operatorname{det}\left[\begin{array}{ccc} 1 & x & x^2 \\ 1 & x^2 & x^4 \\ 1 & x^3 & x^6 \end{array}\right]=0 \text { is equal to }$

Let $A\,\,\mathrm{be}\,\,\mathrm{a}\,3 \times 3\,$ matrix such that $adj\,A\, = \,\left[ {\begin{array}{cccccccccccccccccccc} 2&{ - 1}&1\\ { - 1}&0&2\\ 1&{ - 2}&{ - 1} \end{array}} \right]\,\,\mathrm{and}\,B = \,\,\mathrm{adj}\,\,\mathrm{(adj}\,A\mathrm{)}\,$ $\mathrm{If}\,\left| A \right|\,\mathrm{ = }\,\lambda \,\mathrm{and}\,\left| {{{({B^{ - 1}})}^T}} \right|\,\mathrm{ = }\,\mu,$ then the ordered pair, $\,\left( {\left| \lambda \right|,\,\mu } \right)$ is equal to

The matrix $X$ for which $\left[\begin{array}{cc}1 & -4 \\ 3 & -2\end{array}\right] X=\left[\begin{array}{cc}-16 & -6 \\ 7 & 2\end{array}\right]$ is

Let $A = \left| {\begin{array}{cccccccccccccccccccc} 2&b&1\\ b&{{b^2} + 1}&b\\ 1&b&2 \end{array}} \right|$, where $b > 0$. Then, the minimum value of $\frac{{\det \left( A \right)}}{b}$ is

The determinant $\Delta = \left| {\begin{array}{cccccccccccccccccccc} {{a^2} + {x^2}}&{ab}&{ac}\\ {ab}&{{b^2} + {x^2}}&{bc}\\ {ac}&{bc}&{{c^2} + {x^2}} \end{array}} \right|$ is divisible by

The value of $k \in R$, for which the following system of linear equations $3 x-y+4 z=3, x+2 y-3 z=-2,6 x+5 y+k z=-3$, has infinitely many solutions, is:

If $M$ is square matrix such that $M^2=M$, then $(I+M)^3-7 M$ is equal to

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

Let $P$ be a square matrix such that $P^{2}=I-P$. For $\alpha, \beta, \gamma, \delta \in \mathbb{N},$ if $P^{\alpha}+P^{\beta}=\gamma I-29 P$ and $P^{\alpha}-P^{\beta}=\delta I-13 P$, then $\alpha+\beta+\gamma-\delta$ is equal to

The number of $2 \times 2$ matrices $X$ satistying the matrix equation $X^2=I$ ( $\mathrm{I}$ is $2 \times 2$ unit matrix) is

The matrix $\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]$ is -

Let $A=\left(\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right)$ and $B=\left(\begin{array}{ll}a & 0 \\ 0 & b\end{array}\right), a, b \in N$. Then

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

The determinant $\Delta = \left| {\begin{array}{cccccccccccccccccccc} a&b&{ax + b}\\ b&c&{bx + c}\\ {ax + b}&{bx + c}&0 \end{array}} \right|$ is equal to zero, if

If $\left[\begin{array}{lll}1 & x & 1\end{array}\right]\left[\begin{array}{rrr}1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2\end{array}\right]\left[\begin{array}{l}1 \\ 2 \\ x\end{array}\right]=[0]$ then $x =$

Let $\left|\begin{array}{ccc}x^2+x & 2 x-1 & x+3 \\ 3 x+1 & 2+x^2 & x^3-3 \\ x-3 & x^2+4 & 2 x\end{array}\right|$
$=p x^7+q x^6+r x^5+s x^4+t x^3+u x^2+v x+w \text {, }$
then which of the following is not true?