#self doubt

Question

Answer 1

WRONG APPROACH

we know that $A^{-1} = \frac{1}{|A|} (adj(A))$

Multiply by $A$ on both sides

$AA^{-1} = \frac{1}{|A|}A(adj(A))$ $\Leftrightarrow$ $|A|I = A(adj(A))$

from the matrix given, we conclude that $|A|= \frac{1}{5^{3}}$

We want to find the value of $\vert(5A)^{-1}\vert$.

$\vert(5A)^{-1}\vert =\frac{1}{\vert 5A\vert}=   $  $ \frac{1}{5^{3}\vert A \vert}$ chip in the value of $\vert A\vert$ from above.

Therefore $ \vert (5A)^{-1} \vert = 1$

CORRECT APPROACH

There is a slight in the question , in RHS that is not a determinant but a matrix. So from now on in this solution we will consider it as a matrix.

$A^{-1} = \frac{1}{|A|} (adj(A))$

Multiply by $A^{-1}$ on both sides

$AA^{-1} = \frac{1}{|A|}A(adj(A))$ $\Leftrightarrow$ $|A|I = A(adj(A))$

From the given matrix, we conclude that$|A|I=\frac{1}{5}I \Rightarrow$ $|A|=\frac{1}{5}$

We want to find the value of $\vert(5A)^{-1}\vert$.

$\vert(5A)^{-1}\vert =\frac{1}{\vert 5A\vert}=   $  $ \frac{1}{5^{3}\vert A \vert}$ chip in the value of $\vert A\vert$ from above.

Therefore $ \vert (5A)^{-1} \vert = \frac{1}{5^2}$

Thank you @AGNIDEB+MUKHERJEE

Comments

Ok got it actually I didn't see the determinant I thought A(AdjA)= the matrix is given.😅

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nice observation @AGNIDEB MUKHERJEE .

The question author might have missed something, because now if i think about it

how is $A(adjA)$ giving us a Determinant instead of matrix? It should give us matrix because at the end of the day

$A(adjA)$ is simply a matrix multiplication. 

So yes if we consider RHS as matrix then $|A|=\frac{1}{5}$

 I will update my answer accordingly :)

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I think the author wrote it right but person who asked this question may have misinterpreted matrix and determinant as same thing  and wrote wrong ig.

Answer 2

Simply we can find
A * adj(A) = |A| * I

So, |A| = 1/5..

Now, we know, |(A)^-1| = (1/5)^-1 = 5..

So, |(5*A)^(-1)| = (1/5)^(3) * 5 = 1/125 * 5 = 1/25... { We're applying (1/5)^3 as there are 3 rows and 5 will be multiplied to each of the rows....}