For a real number $\alpha$, if the system $\left[ {\begin{array}{*{20}{r}} 1&\alpha &{{\alpha ^2}} \\ \alpha &1&\alpha \\ {{\alpha ^2}}&\alpha &1 \end{array}} \right]\left[ {\begin{array}{*{20}{l}} x \\ y \\ z \end{array}} \right] = \left[ {\begin{array}{*{20}{r}} 1 \\ { - 1} \\ 1 \end{array}} \right]$ of the linear equations, has infinitely many solutions, then $1+\alpha+\alpha^2=$

JEE MATHEMATICS MCQ: MATRICES

1 Easy

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Analyzing +4 | -1

For a real number

\alpha

, if the system

\left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr} 1&\alpha &{{\alpha ^2}} \\ \alpha &1&\alpha \\ {{\alpha ^2}}&\alpha &1 \end{array}} \right]\left[ {\begin{array}{llllllllllllllllllll} x \\ y \\ z \end{array}} \right] = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr} 1 \\ { - 1} \\ 1 \end{array}} \right]

of the linear equations, has infinitely many solutions, then

1+\alpha+\alpha^2=

View answer and explanation

Correct option: A

1

Option A is correct because when

\alpha = -1

, the system has infinitely many solutions, and for this value,

1 + \alpha + \alpha^2 = 1

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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|