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 options is are always TRUE-A- M - IB- detM-1C- M 2- ID- adj M 2- I

The correct options are B det(M)=1 C M^2=I nbsp; D (adj M)^2=IGiven, M^ 1=adj(adj M) ⇒ |M^ 1|=|adj(adj M)| nbsp; ⇒1|M|=|M|^(n 1)^2 ( ∵ |adj(adj A)=| ...

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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 $M^{- 1} = a d j (a d j M),$ then which of the following options is (are) always TRUE?

M = I

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d e t (M) = 1

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M^{2} = I

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(a d j M)^{2} = I

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Solution

The correct options are
B $d e t (M) = 1$
C $M^{2} = I$
D $(a d j M)^{2} = I$
Given, $M^{- 1} = a d j (a d j M)$
$\Rightarrow | M^{- 1} | = | a d j (a d j M) |$
$\Rightarrow \frac{1}{| M |} = | M |^{(n - 1)^{2}} (∵ | a d j (a d j A) = | A |^{(n - 1)^{2}})$

For $n = 3$
$\frac{1}{| M |} = | M |^{4}$
$\Rightarrow | M |^{5} = 1$
$\Rightarrow | M | = 1 \dots (1)$

Now, $M^{- 1} = a d j (a d j M)$
$\Rightarrow M^{- 1} = | M |^{3 - 2} \cdot M (∵ a d j (a d j A) = | A |^{n - 2} \cdot A)$
$\Rightarrow M^{- 1} = | M | \cdot M$
$\Rightarrow M^{- 1} = M \dots$ [from equation $(1)$ ]
$\Rightarrow M \times M^{- 1} = M \times M$
$\Rightarrow I = M^{2}$
or, $M^{2} = I$

Now, multiply $a d j (M)$ both sides
$a d j (M) \cdot M \cdot M = a d j (M) \cdot I$
Since $A \cdot a d j (A) = | A | \cdot I,$ we have
$(| M | \cdot I) \cdot M = a d j (M)$
$\Rightarrow I \cdot M = a d j (M) \dots (∵ | M | = 1)$
$\Rightarrow M = a d j (M)$

Again, multiply $a d j (M)$
$M \cdot a d j (M) = (a d j (M))^{2}$
$\Rightarrow | M | \cdot I = (a d j (M))^{2}$
$\Rightarrow I = (a d j (M))^{2}$

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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 options is (are) always TRUE?

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