If left| {,begin{array}{*{20}{c}}a&b&00&a&bb&0&aend{array},} right| = 0 , th

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3 and 4 .Determinants and Matrices

easy

If $\left| {\,\begin{array}{*{20}{c}}a&b&0\\0&a&b\\b&0&a\end{array}\,} \right| = 0$, then

$a $ is one of the cube roots of unity

$b$ is one of the cube roots of unity

$\left( {\frac{a}{b}} \right)$is one of the cube roots of unity

$\left( {\frac{a}{b}} \right)$is one of the cube roots of $ -1$

Solution

(d) Given, $\Delta = \left| {\,\begin{array}{*{20}{c}}a&b&0\\0&a&b\\b&0&a\end{array}\,} \right|\, = \,0.$
Expanding the given determinant, we get $a({a^2} – 0) – b(0 – {b^2}) = 0$ or ${a^3} + {b^3} = 0.$
This equation may be written as ${\left( {\frac{a}{b}} \right)^3} = – 1.$
Therefore, $\left( {\frac{a}{b}} \right)$ is one of the cube roots of $ -1$.

Standard 12

Mathematics

If $\left|\begin{array}{cc}x & 2 \\ 18 & x\end{array}\right|=\left|\begin{array}{cc}6 & 2 \\ 18 & 6\end{array}\right|,$ then $x$ is equal to

easy

$\left| {\,\begin{array}{*{20}{c}}1&1&1\\a&b&c\\{{a^3}}&{{b^3}}&{{c^3}}\end{array}\,} \right| = $

medium

If ${x^a}{y^b} = {e^m},{x^c}{y^d} = {e^n},{\Delta _1} = \left| {\,\begin{array}{{20}{c}}m&b\\n&d\end{array}\,} \right|\,\,{\Delta _2} = \left| {\,\begin{array}{{20}{c}}a&m\\c&n\end{array}\,} \right|$ and ${\Delta _3} = \left| {\,\begin{array}{*{20}{c}}a&b\\c&d\end{array}\,} \right|$, then the values of $x$ and $y$ are respectively

hard

If the system of linear equations

$2 x+y-z=3$

$x-y-z=\alpha$

$3 x+3 y+\beta z=3$

has infinitely many solution, then $\alpha+\beta-\alpha \beta$ is equal to …. .

medium

(JEE MAIN-2021)

If area of triangle is $35$ $\mathrm{sq}$ $\mathrm{units}$ with vertices $(2,-6),(5,4)$ and $(\mathrm{k}, 4) .$ Then $\mathrm{k}$ is

medium

If $\left| {\,\begin{array}{*{20}{c}}a&b&0\\0&a&b\\b&0&a\end{array}\,} \right| = 0$, then

Solution

Similar Questions

If $\left|\begin{array}{cc}x & 2 \\ 18 & x\end{array}\right|=\left|\begin{array}{cc}6 & 2 \\ 18 & 6\end{array}\right|,$ then $x$ is equal to

$\left| {\,\begin{array}{*{20}{c}}1&1&1\\a&b&c\\{{a^3}}&{{b^3}}&{{c^3}}\end{array}\,} \right| = $

If the system of linear equations $2 x+y-z=3$ $x-y-z=\alpha$ $3 x+3 y+\beta z=3$ has infinitely many solution, then $\alpha+\beta-\alpha \beta$ is equal to …. .

If area of triangle is $35$ $\mathrm{sq}$ $\mathrm{units}$ with vertices $(2,-6),(5,4)$ and $(\mathrm{k}, 4) .$ Then $\mathrm{k}$ is

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If the system of linear equations

$2 x+y-z=3$

$x-y-z=\alpha$

$3 x+3 y+\beta z=3$

has infinitely many solution, then $\alpha+\beta-\alpha \beta$ is equal to …. .