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STD 12 - 3 and 4 . determinant and metrices — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 3 and 4 . determinant and metrices1 Mark
MCQ
Let $A=\left[\begin{array}{cc}0 & -2 \\ 2 & 0\end{array}\right]$. If $M$ and $N$ are two matrices given by $M =\sum \limits_{ k =1}^{10} A ^{2 k }$ and $N =\sum \limits_{ k =1}^{10} A ^{2 k -1}$ then $MN ^{2}$ is
  • ✓
    a non-identity symmetric matrix
  • B
    skew symmetric matrix
  • C
    neither symmetric nor skew-symmetric matrix
  • D
    an identify matrix
✓

Answer

Correct option: A.
a non-identity symmetric matrix
a
$A =\left[\begin{array}{cc}0 & -2 \\ 2 & 0\end{array}\right]$

$A ^{2}=\left[\begin{array}{cc}0 & -2 \\ 2 & 0\end{array}\right]\left[\begin{array}{cc}0 & -2 \\ 2 & 0\end{array}\right]=\left[\begin{array}{cc}-4 & 0 \\ 0 & -4\end{array}\right]=-4 I$

$A ^{3}=-4 A$

$A ^{4}=(-4 I )(-4 I )=(-4)^{2} I$

$A ^{5}=(-4)^{2} A , A ^{6}=(-4)^{3} I$

$M=\sum \limits_{ k =1}^{10} A ^{2 k }= A ^{2}+ A ^{4}+\ldots .+ A ^{20}$

$=\left[-4+(-4)^{2}+(-4)^{3}+\ldots+(-4)^{20}\right] I$

$=-4 \lambda I$

$\Rightarrow \quad M$ is symmetric matrix

$N =\sum \limits_{ k =1}^{10} A ^{2 k -1}= A + A ^{3}+\ldots \ldots+ A ^{19}$

$= A \left[1+(-4)+(-4)^{2}+\ldots .+(-4)^{9}\right]$

$=\lambda A \Rightarrow \text { skew symmetric }$

$\Rightarrow N ^{2} \text { is symmetric matrix }$

$\Rightarrow MN ^{2}$ is non identity symmetric matrix

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