If $A = \left[ {\begin{array}{*{20}{c}} 1&{ - 1}&1\\ 0&2&{ - 3}\\ 2&1&0 \end{array}} \right]$ and $\;B = \left( {{\rm{adj}}\;A} \right),\;$ and $C = 5A,\;$ then $\;\frac{{\left| {{\rm{adj}}\;B} \right|}}{{\left| C \right|}}\;$ is equal to

JEE MATHEMATICS MCQ: MATRICES

1 Medium

1 of 1

Application +4 | -1

A = \left[ {\begin{array}{cccccccccccccccccccc} 1&{ - 1}&1\\ 0&2&{ - 3}\\ 2&1&0 \end{array}} \right]

and

\;B = \left( {\mathrm{adj}\;A} \right),\;

and

C = 5A,\;

then

\;\frac{{\left| {\mathrm{adj}\;B} \right|}}{{\left| C \right|}}\;

is equal to

View answer and explanation

Correct option: D

Option D (1) is correct because the ratio simplifies to 1 after applying properties of adjoint and determinants.

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\left|\begin{array}{ccc}a & b-c & b+c \\ c+a & b & c-a \\ a-b & a+b & c\end{array}\right|=0

a, b, c

a x+b y+c=0

{x_2} - {x_3} = 1,\; - {x_1} + 2{x_3} = - 2,\;{x_1} - 2,\;{x_1} - 2{x_2} = 3

2x + 3y + 1 = 0,3x + y - 2 = 0

ax + 2y - b = 0

\left|\begin{array}{ccc}1 & \left(\alpha^{2 x}-\alpha^{-2 x}\right)^2 & \left(\alpha^{2 x}+\alpha^{-2 x}\right)^2 \\ 1 & \left(\beta^{2 x}-\beta^{-2 x}\right)^2 & \left(\beta^{2 x}+\beta^{-2 x}\right)^2 \\ 1 & \left(\gamma^{2 x}-\gamma^{-2 x}\right)^2 & \left(\gamma^{2 x}+\gamma^{-2 x}\right)^2\end{array}\right|

\omega

\left| {\begin{array}{cccccccccccccccccccc} 1&\omega &{{\omega ^2}}\\ \omega &{{\omega ^2}}&1\\ {{\omega ^2}}&1&\omega \end{array}} \right|

A

2 \times 2

A(\operatorname{adj} A)=\left[\begin{array}{cc}10 & 0 \\ 0 & 10\end{array}\right]

|A|