If A is a matrix [ 1 -1 0 0 2 a 1 1 2 ] , then the number of real value (.s.) of 'a' for which A(.adjA.)=adj(.adjA.), is (where adjA denotes adjoint of matrix A )

Question

If A is a matrix [ 1 -1 0 0 2 a 1 1 2 ] , then the number of real value (.s.) of 'a' for which A(.adjA.)=adj(.adjA.), is (where adjA denotes adjoint of matrix A ) (A) 0 (B) 1 (C) 2 (D) 3

Tardigrade Admin · Accepted Answer

|.A|.=4-a+1(.-a.)=4-2a A(.AdjA.)=|.A|.I=(.4-2a.)I Adj(.AdjA.)=|.A|.A=(.4-2a.)A (.4-2a.)I=(.4-2a.)A ⇒ (.4-2a.)(.A-I.)=0⇒ a=2

Q. If $A$ is a matrix $⎣ ⎡ 101 - 1 21 0 a 2 ⎦ ⎤$ , then the number of real value $(s)$ of $^{'} a^{'}$ for which $A (a d j A) = a d j (a d j A),$ is (where $a d j A$ denotes adjoint of matrix $A$ )

$0$

$1$

$2$

$3$

$∣ A ∣ = 4 - a + 1 (- a) = 4 - 2 a$
$A (A d j A) = ∣ A ∣ I = (4 - 2 a) I$
$A d j (A d j A) = ∣ A ∣ A = (4 - 2 a) A$
$(4 - 2 a) I = (4 - 2 a) A$
$\Rightarrow (4 - 2 a) (A - I) = 0 \Rightarrow a = 2$

Q. If A is a matrix ⎣⎡​101​−121​0a2​⎦⎤​ , then the number of real value (s) of ′a′ for which A(adjA)=adj(adjA), is (where adjA denotes adjoint of matrix A )

0

1

2

3

∣A∣=4−a+1(−a)=4−2a A(AdjA)=∣A∣I=(4−2a)I Adj(AdjA)=∣A∣A=(4−2a)A (4−2a)I=(4−2a)A ⇒(4−2a)(A−I)=0⇒a=2

Q. If $A$ is a matrix $⎣ ⎡ 101 - 1 21 0 a 2 ⎦ ⎤$ , then the number of real value $(s)$ of $^{'} a^{'}$ for which $A (a d j A) = a d j (a d j A),$ is (where $a d j A$ denotes adjoint of matrix $A$ )

$0$

$1$

$2$

$3$

$∣ A ∣ = 4 - a + 1 (- a) = 4 - 2 a$
$A (A d j A) = ∣ A ∣ I = (4 - 2 a) I$
$A d j (A d j A) = ∣ A ∣ A = (4 - 2 a) A$
$(4 - 2 a) I = (4 - 2 a) A$
$\Rightarrow (4 - 2 a) (A - I) = 0 \Rightarrow a = 2$