Let A= [ cc 0 & -2 2 & 0 ]. If M and N are two matrices given by M = _ k =1 ^ 10 A ^ 2 k and N = _ k =1 ^ 10 A ^ 2 k -1 then MN ^ 2 is

Let A= [ cc 0 & -2 2 & 0 ]. If M and N are two matrices given by …

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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