Question

A is 3 x 3 metrics and |adj| = 4 then find

(i) |2A|

(ii) |-3A^-1|

(iii) A.AdjA 

Answer 1

A is matrix of order 3.

\(|adjA| = 4\)

\(\because |adjA| = |A|^{3 - 1}= |A|^2\)

\(\therefore |A|^2 = 4\)

⇒ \(|A| = \pm 2\)

(i) \(|2A| = 2^3|A|= 8(\pm2) = \pm 16\)

(ii) \(|-3A^{-1}| = (-3)^3 |A^{-1}| = \frac{-27}{|A|} = \frac{-27}{\pm2}= \pm\frac{27}2\)

(iii) \(A.adj A = |A|I= \pm 2I= \begin{bmatrix}\pm2&0&0\\0&\pm2&0\\0&0&\pm2\end{bmatrix}\)