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  • Let <img alt="A=\left[\begin{array}{rr} 0 & -2 \\ 2 & 0 \end{array}\right]" src="https://entrancecorner.oncodecogs.com/gif.latex?A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Brr%7D%200%20%26%20-

Let A=\left[\begin{array}{rr} 0 & -2 \\ 2 & 0 \end{array}\right]. If M and N are two matrices given by M=\sum_{k=1}^{10} A^{2 k} \text { and } N=\sum_{k=1}^{10} A^{2 k-1}. Then \mathrm{MN}^{2} is:

Option: 1

a non-identity symmetric matrix


Option: 2

a skew-symmetric matrix


Option: 3

neither symmetric nor skew-symmetric matrix


Option: 4

an identity matrix


Answers (1)

best_answer

\mathrm{A^{2}= \begin{bmatrix} 0 &-2 \\ 2& 0 \end{bmatrix}\begin{bmatrix} 0 &-2 \\ 2& 0 \end{bmatrix}= \begin{bmatrix} -4 &0 \\ 0& -4 \end{bmatrix}= -4I\, : symmetric}
\mathrm{\& \: A^{3}=-4 A( Skew\: Symmetric)}

\mathrm{\Rightarrow M=\sum_{k=1}^{10} A^{2 k}=\left[(-4)+(-4)^{2}+(-4)^{3}+\cdots +\left ( -4 \right )^{10}\right]\, I}
                                 \mathrm{=-4 \: \lambda \: I \text { is Symmetric } }

\mathrm{\Rightarrow N =\sum_{k=1}^{10} A^{2 k-1}=A\left[1+(-4)+(-4)^{3}+\cdots+(-4)^{9}\right] I }
                               \mathrm{=\lambda \: A\: \: is \: Skew \: Symmetric}
\mathrm{where \: \: \lambda=\left\{1+(-4)+(-4)^{3}+\cdots \cdot+(-4)^{9}\right\} }

MN^2=-4 \lambda^3 A^2
\mathrm{\Rightarrow M N^{2}\text{ is Symmetric matrix}}

Posted by

himanshu.meshram

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