Let M be a 3×3 invertible matrix with real entries and let I denote the 3×3 identity matrix. M −1 =adj adj M , then which of the following statements is/are ALWAYS TRUE?

JEE Advanced · Mathematics · 18. Matrices

Let $M$ be a $3 \times 3$ invertible matrix with real entries and let $I$ denote the $3 \times 3$ identity matrix. $M^{- 1} = a d j (a d j M)$ , then which of the following statements is/are ALWAYS TRUE?

A $M = I$
B $\det M = 1$
C $M^{2} = I$
D ${(a d j M)}^{2} = I$

Verified Solution

Answer & Solution

Correct Answer

(B) $\det M = 1$

Step-by-step Solution

Detailed explanation

Given

M^{- 1} = adj (adjM)

∴ adj M \cdot M^{- 1} = adj M . adj (adjM)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

Column 1	Column 2	Column 3
(I) $x^{2} + y^{2} = a^{2}$	(i) $m y = m^{2} x + a$	(P) $(\frac{a}{m^{2}}, \frac{2 a}{m})$
(II) $x^{2} + a^{2} y^{2} = a^{2}$	(ii) $y = m x + a \sqrt{m^{2} + 1}$	(Q) $(\frac{- m a}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}})$
(III) $y^{2} = 4 a x$	(iii) $y = m x + \sqrt{a^{2} m^{2} - 1}$	(R) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{1}{\sqrt{a^{2} m^{2} + 1}})$
(IV) $x^{2} - a^{2} y^{2} = a^{2}$	(iv) $y = m x + \sqrt{a^{2} m^{2} + 1}$	(S) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} - 1}}, \frac{- 1}{\sqrt{a^{2} m^{2} - 1}})$

The tangent to a suitable conic (Column 1) at

(\sqrt{3}, \frac{1}{2})

is found to be

\sqrt{3} x + 2 y = 4

, then which of the following options is the only Correct combination?

JEE Advanced 2017 Hard

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