Question

If A is an invertible matrix of order 3 and |Adj A| = 25, then write the value of |5A- 1|.​

Answer 1

$\textbf{Given:}$

$\textsf{A is an invertible matrix order 3 and}$

$\mathsf{|adjA|=25}$

$\textbf{To find:}$

$\textsf{The value of}\;\mathsf{|5\,A^{-1}|}$

$\textbf{Solution:}$

$\textbf{Formula used:}$

$\boxed{\begin{minipage}{7cm} \\\textsf{Let A be a square matrix of order n}\\\\\mathsf{Then,}\\\\\mathsf{1.\;|adjA|=|A|^{n-1}}\\\\\mathsf{2.\;|k\,A|=k^{n}|A|}\\\\\mathsf{3.\;|A^{-1}|=\dfrac{1}{|A|}}\\ \end{minipage}}$

$\mathsf{Consider,}$

$\mathsf{|adjA|=25}$

$\implies\mathsf{|A|^2=25}$

$\implies\mathsf{|A|=\sqrt{25}}$

$\implies\mathsf{|A|=5}$

$\textsf{Now,}$

$\mathsf{|5\,A^{-1}|}$

$\textsf{Using formula (2),}$

$\mathsf{=5^3\,|A^{-1}|}$

$\mathsf{=125{\times}\dfrac{1}{|A|}}$

$\mathsf{=125{\times}\dfrac{1}{5}}$

$\mathsf{=25}$

$\implies\boxed{\mathsf{|5\,A^{-1}|=25}}$

$\textbf{Find more:}$

1.A matrix of order 3X3 has determinant 5. What is the value of |3A|?

https://brainly.in/question/3052917

2.A is a square matrix with |A|=4 then find the value of |A. (adjA)|

https://brainly.in/question/10368771