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Determinants — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDeterminants1 Mark
MCQ
Assertion (A) : If $A=\left(\begin{array}{ccc}l & 0 & 0 \\ 0 & m & 0 \\ 0 & 0 & n\end{array}\right)$, then
$
A^{-1}=\left(\begin{array}{ccc}
1 / l & 0 & 0 \\
0 & 1 / m & 0 \\
0 & 0 & 1 / n
\end{array}\right)
$
Reason $( R )$ : The inverse of a diagonal matrix is a diagonal matrix.
  • A
    Both (A) and (R) are true and (R) is the correct explanation of (A).
  • ✓
    Both (A) and (R) are true but (R) is not the correct explanation of (A).
  • C
    (A) is true but (R) is false.
  • D
    (A) is false but (R) is true.
✓

Answer

Correct option: B.
Both (A) and (R) are true but (R) is not the correct explanation of (A).
(b) : $\because A=\left(\begin{array}{ccc}l & 0 & 0 \\ 0 & m & 0 \\ 0 & 0 & n\end{array}\right)$
$\therefore \quad|A|=l m n$ and $\operatorname{adj}(A)=\left(\begin{array}{ccc}m n & 0 & 0 \\ 0 & l n & 0 \\ 0 & 0 & l m\end{array}\right)$
$
\therefore \quad A^{-1}=\frac{\operatorname{adj} A}{|A|}=\left(\begin{array}{ccc}
1 / l & 0 & 0 \\
0 & 1 / m & 0 \\
0 & 0 & 1 / n
\end{array}\right)=\operatorname{diag}\left(\frac{1}{l}, \frac{1}{m}, \frac{1}{n}\right)
$
$\therefore \quad$ Assertion and reason are both correct but reason is not the correct explanation of assertion.

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