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  • Fill in the blanks in each of the If A is a symmetric matrix, then A3 is a ______ matrix.

Fill in the blanks in each of the
If A is a symmetric matrix, then A^3 is a ______ matrix.

 

Answers (1)

A\textsuperscript{3} is Also a symmetric matrix.

We are given that: A’=A ..(1)

\\ \Rightarrow (A\textsuperscript{2})'=(AA)'=A'A' \\ \Rightarrow A'A'=(A)(A)=A\textsuperscript{2} \\ \Rightarrow (A\textsuperscript{2})'=A\textsuperscript{2} (symmetric matrix) ..(2) \\ \Rightarrow (A\textsuperscript{3})'=(A(A\textsuperscript{2}))'=(A\textsuperscript{2})'A' \\ \Rightarrow (A\textsuperscript{2})'A'=A\textsuperscript{2}A= A\textsuperscript{3} (Using (1) and (2) ) \\ \Rightarrow (A\textsuperscript{3})'=A\textsuperscript{3} (symmetric matrix)

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