\( M \) is a square matrix of order 3. If \( M(\mathrm{adj}\,M)=\begin{bmatrix}5 & 0 & 0\\0 & 5 & 0\\0 & 0 & 5\end{bmatrix} \), then the value of \( |M|+|\mathrm{adj}\,M| \) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 30
Since \( M\,(\mathrm{adj}\,M)=|M|\,I \) and the given matrix is \( 5I \), |M| = 5. For a matrix of order 3, \( |\mathrm{adj}\,M|=|M|^{3-1} \):
\( \displaystyle |\mathrm{adj}\,M|=5^{2}=25 \)
\( \displaystyle |M|+|\mathrm{adj}\,M|=5+25 \)
\( \displaystyle =30 \)
Why the other options are incorrect
Option (1) 125 — Incorrect. 125 is the determinant of the given matrix 5I (that is, \( |M|^3 \)), not the required sum.
Option (3) 25 — Incorrect. 25 is \( |\mathrm{adj}\,M| \) alone; |M| = 5 must be added to it.
Option (4) 50 — Incorrect. The sum is 5 + 25 = 30, not 50.