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

(B) $\det M=1$; (C) $M^{2}=I$; (D) $(\text{adj } M)^{2}=I$

JEE Advanced · 2020 · 27 Sep 2020 · Paper 1 · Matrices

Let M be a

3 \times 3

invertible matrix with real entries and let I denote the

3 \times 3

identity matrix. If

M^{- 1} = adj (adj M)

, then which of the following statement is/are ALWAYS TRUE?

Answer: (B), (C), (D)

We know that

adj (adj M) = (det M)^{n - 2} M

for an

n \times n

matrix. For

n = 3

adj (adj M) = (det M) M

.

Given

M^{- 1} = adj (adj M) = (det M) M

.

Multiplying both sides by

M

I = (det M) M^{2}

.

Taking determinants:

1 = (det M)^{3}

, so

det M = 1

. (B) is TRUE.

From

I = (det M) M^{2} = M^{2}

, we have

M^{2} = I

. (C) is TRUE.

Also,

(adj M)^{2} = adj (M^{2}) = adj (I) = I

. (D) is TRUE.

(A) is FALSE because

M^{2} = I

does not imply

M = I

(e.g.,

M = - I

also satisfies

M^{2} = I

JEE Advanced · 2020 · 27 Sep 2020 · Paper 1 · One or more correct options