Question

If A
is Involuntary matrix then
1/2 ( I + A ) is .. ( with fully solved solution)

1) Idenpoilent Matrix
2) Nilpolent Matrix
3) Periodic matrix
4) involutary matrix
​

Answer 1

$\textbf{Given:}$

$\textsf{A is Involutory matrix}$

$\textbf{To find:}$

$\mathsf{Type\;of\;the\;matrix\;\dfrac{1}{2}(I+A)}$

$\textbf{Solution:}$

$\textbf{Concept used:}$

$\textbf{A square matrix A is said to Involutory if}\;\mathsf{A^2=I}$

$\textbf{A square matrix A is said to Idempotent if}\;\;\mathrm{A^2=A}$

$\textsf{Since A is involutory,}\;\;\mathrm{A^2=I}$

$\textsf{Consider,}$

$\mathsf{\dfrac{1}{2}(I+A){\times}\dfrac{1}{2}(I+A)}$

$\mathsf{=\dfrac{1}{4}(I+A)^2}$

$\mathsf{=\dfrac{1}{4}(I^2+A^2+2\,A\,I)}$

$\mathsf{=\dfrac{1}{4}(I+I+2\,A\,I)}$

$\mathsf{=\dfrac{1}{4}(2I+2\,A\,I)}$

$\mathsf{=\dfrac{1}{4}{\times}2(I+A\,I)}$

$\mathsf{=\dfrac{1}{2}(I+A)}$

$\mathsf{\dfrac{1}{2}(I+A){\times}\dfrac{1}{2}(I+A)=\dfrac{1}{2}(I+A)}$

$\therefore\textsf{A is idempotent}$

$\textbf{Answer:}$

$\textsf{Option (1) is correct}$

Answer 2

Answer:

Step-by-step explanation:

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