JEE Advance - Mathematics (2020 - Paper 1 Offline - No. 8)
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
det M = 1
M2 = I
(adj M)2 = I
Explanation
It is given that matrix M be a 3 $$ \times $$ 3 invertible matrix, such that
M$$-$$1 = adj(adj M) $$ \Rightarrow $$ M$$-$$1 = |M| M
($$ \because $$ for a matrix A of order 'n' adj(adjA) = |A|n$$-$$2 A}
$$ \Rightarrow $$ M$$-$$1 M = |M|M2 $$ \Rightarrow $$ M2|M| = I .....(i)
$$ \because $$ det(M2 |M|) = det(I) = 1
$$ \Rightarrow $$ |M|3|M|2 = 1 $$ \Rightarrow $$ |M| = 1 .....(ii)
from Eqs. (i) and (ii), we get
M2 = I
As, adj M = |M|M$$-$$1 = M
$$ \Rightarrow $$ (adj M)2 = M2 (adj M)2 = I
M$$-$$1 = adj(adj M) $$ \Rightarrow $$ M$$-$$1 = |M| M
($$ \because $$ for a matrix A of order 'n' adj(adjA) = |A|n$$-$$2 A}
$$ \Rightarrow $$ M$$-$$1 M = |M|M2 $$ \Rightarrow $$ M2|M| = I .....(i)
$$ \because $$ det(M2 |M|) = det(I) = 1
$$ \Rightarrow $$ |M|3|M|2 = 1 $$ \Rightarrow $$ |M| = 1 .....(ii)
from Eqs. (i) and (ii), we get
M2 = I
As, adj M = |M|M$$-$$1 = M
$$ \Rightarrow $$ (adj M)2 = M2 (adj M)2 = I
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