IIT JEE Main Maths -Unit 3 -Determinants- Exam Style Questions- New Syllabus

Question

Let $ A $ be a square matrix of order 3 such that $ \det(A) = -2 $ and $ \det(3 \text{adj}(-6 \text{adj}(3A))) = 2^{m+n} \times 3^{mn} $, $ m > n $. Then $ 4m + 2n $ is equal to _____.

▶️ Answer/Explanation

Ans. (34)

Sol. $ |A| = -2 $

$ \det(3 \text{adj}(-6 \text{adj}(3A))) = 3^3 \det(\text{adj}(-6 \text{adj}(3A))) $

$ = 3^3 (-6)^6 (\det(3A))^4 $

$ = 3^{21} \times 2^{10} $

$ m + n = 10 $

$ mn = 21 $

$ m = 7, n = 3 $

$ 4m + 2n = 4 \times 7 + 2 \times 3 = 28 + 6 = 34 $

Question

For a $3 \times 3$ matrix $M$, let trace $(M)$ denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A| = \frac{1}{2}$ and trace $(A) = 3$. If $B = \text{adj}(\text{adj}(2A))$, then the value of $|B| + \text{trace}(B)$ equals:
(1) 56
(2) 132
(3) 174
(4) 280

▶ Answer/Explanation

Ans. (4)
Sol. $|A|= \frac{1}{2}$, trace$(A) = 3$, $B = \text{adj}(\text{adj}(2A))=|2A|^{n-2}(2A)$
$n = 3$, $B = |2A|(2A) = 2^3.|A|(2A) = 8A$
$|B| = |8A| = 8^3.|A| = 2^8 = 256$
trace$(B) = 8 \text{ trace}(A) = 24$
$|B| + \text{trace}(B) = 280$

Question

If A, B and $(\text{adj}(A^{-1}) + \text{adj}(B^{-1}))$ are non-singular matrices of same order, then the inverse of $A(\text{adj}(A^{-1}) + \text{adj}(B^{-1}))^{-1}B$, is equal to

(1) $AB^{-1} + A^{-1}B$
(2) $\text{adj}(B^{-1}) + \text{adj}(A^{-1})$
(3) $AB^{-1} (\text{adj}(B) + \text{adj}(A))$
(4) $\frac{1}{AB} (A^{-1}B + BA^{-1})$

▶️ Answer/Explanation

Ans. (3)

Sol.

$\left[ A(\text{adj}(A^{-1}) + \text{adj}(B^{-1})) \right] B^{-1}$

$B^{-1}(\text{adj}(A^{-1}) + \text{adj}(B^{-1})) A^{-1}$

$B^{-1}\text{adj}(A^{-1}) A^{-1} + B^{-1}(\text{adj}(B^{-1})) A^{-1}$

$B^{-1} |A^{-1}| I + B^{-1} |A^{-1}|$

$\frac{B^{-1}}{|A|} + \frac{A^{-1}}{|B|}$

$\Rightarrow \frac{\text{adj}B}{|B| |A|} + \frac{\text{adj}A}{|A| |B|}$

$= \frac{1}{|A| |B|} (\text{adj}B + \text{adj}A)$

IIT JEE Main Maths -Unit 3 -Determinants- Exam Style Questions- New Syllabus

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