X - 0 - 0 10 - y- 00 - 0 - Za - 0 - 0 10 - b - 0 10 - 0 - Clend matrix -right - then prove that PQ- left - matrix xa - 0 - 0 10 - y b - 0 10 - 0 - ZClend matrix -right - QP

PQ= [ x 0 0 0 y 0 0 0 z ] and Q= [ a 0 0 0 b 0 0 0 c ]= [ xa 0 0 0 yb 0 0 0 zc ] ...(i) and QP = [ a 0 ...

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Inverse of a Matrix

X ∼∼ 0 ∼∼ 0 1...

Question

If P= $⎡ ⎢ ⎣ \begin{matrix} x 00 0 y 0 00 z \end{matrix} ⎤ ⎥ ⎦ a n d Q = ⎡ ⎢ ⎣ \begin{matrix} a 00 0 b 0 00 c \end{matrix} ⎤ ⎥ ⎦$ then prove that

$P Q = ⎡ ⎢ ⎣ \begin{matrix} x a 00 0 y b 0 00 z c \end{matrix} ⎤ ⎥ ⎦ = Q P$

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Solution

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If x,y, z are non-zero real numbers, then the inverse of matrix

$A = ⎡ ⎢ ⎣ \begin{matrix} x & 0 & 0 0 & y & 0 0 & 0 & z \end{matrix} ⎤ ⎥ ⎦$ is

a) $⎡ ⎢ ⎣ \begin{matrix} x^{- 1} & 0 & 0 0 & y^{- 1} & 0 0 & 0 & z^{- 1} \end{matrix} ⎤ ⎥ ⎦$

b) $x y z ⎡ ⎢ ⎣ \begin{matrix} x^{- 1} & 0 & 0 0 & y^{- 1} & 0 0 & 0 & z^{- 1} \end{matrix} ⎤ ⎥ ⎦$

c) $\frac{1}{x y z} ⎡ ⎢ ⎣ \begin{matrix} x & 0 & 0 0 & y & 0 0 & 0 & z \end{matrix} ⎤ ⎥ ⎦$

d) $\frac{1}{x y z} ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦$

A = ⎡ ⎢ ⎣ \begin{matrix} a & 0 & 0 0 & b & 0 0 & 0 & c \end{matrix} ⎤ ⎥ ⎦

A^{n} =

⎧ ⎨ ⎩ \begin{matrix} \frac{x . g (x)}{| x |}, & x \neq 0 0, & x = 0 \end{matrix}

If $A = ⎡ ⎢ ⎣ \begin{matrix} a & 0 & 0 0 & b & 0 0 & 0 & c \end{matrix} ⎤ ⎥ ⎦$ , then $A^{n} =$

A = ⎡ ⎢ ⎣ \begin{matrix} a & 0 & 0 0 & b & 0 0 & 0 & c \end{matrix} ⎤ ⎥ ⎦

A^{n} =

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