Let M be a 3- 3 invertible matrix with real entries and let I denote the 3- 3 identity matrix- If M-1-adjadj M- then which of the following statements is-are ALWAYS TRUE-

M^ 1=adj (adj (M))⋯(1) and M^ 1=1|M|adj(M)⋯(2) Now from (1) and (2) 1|M||M|^2=|M|^4 ⇒ |M|=1 From (2) M^ 1=adj(M)⋯(3) Using (1) M=adj(M)⋯(4) From (3) and (4) I=( ...

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Byju's Answer

Standard XII

Mathematics

Determinant

Let M be a 3×...

Question

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 statements is/are ALWAYS TRUE?

M = I

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det M = 1

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M^{2} = I

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(adj M)^{2} = I

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Solution

The correct option is D $(adj M)^{2} = I$
$M^{- 1} = adj (adj (M)) \dots (1)$
and $M^{- 1} = \frac{1}{| M |} adj (M) \dots (2)$
Now from $(1)$ and $(2)$
$\frac{1}{| M |} | M |^{2} = | M |^{4} \Rightarrow | M | = 1$
From $(2)$
$M^{- 1} = adj (M) \dots (3)$
Using $(1)$
$M = adj (M) \dots (4)$
From $(3)$ and $(4)$
$I = {(adj (M))}^{2} = (M)^{2}$

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

The correct option is D (adj M)2=IM−1=adj (adj (M))⋯(1) and M−1=1|M|adj(M)⋯(2) Now from (1) and (2) 1|M||M|2=|M|4⇒|M|=1 From (2) M−1=adj(M)⋯(3) Using (1) M=adj(M)⋯(4) From (3) and (4) I=(adj(M))2=(M)2

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 statements is/are ALWAYS TRUE?