If A=[ 2 2 9 4 ] and A2+aA+bΙ=O , then a+2b is equal to (where, Ι is an identity matrix and O is a null matrix of order 2 respectively)

Question

If A=[ 2 2 9 4 ] and A2+aA+bΙ=O , then a+2b is equal to (where, Ι is an identity matrix and O is a null matrix of order 2 respectively) (A) 27 (B) -26 (C) 24 (D) 12

Tardigrade Admin · Accepted Answer

A=[ 2 2 9 4 ]⇒ A2=[ 2 2 9 4 ][ 2 2 9 4 ]=[ 22 12 54 34 ] aA=[ 2a 2a 9a 4a ],bΙ=[ b 0 0 b ] A2+aA+bΙ=[ 22+2a+b 12+2a 9a+54 34+4a+b ]=O ⇒ a=-6 ⇒ 22+2a+b=0⇒ b=-10 ⇒ a+2b=-26

Q. If $A = [2924]$ and $A^{2} + a A + b I = O$ , then $a + 2 b$ is equal to (where, $I$ is an identity matrix and $O$ is a null matrix of order $2$ respectively)

$27$

$- 26$

$24$

$12$

$A = [2924] \Rightarrow A^{2} = [2924] [2924] = [22541234]$
$a A = [2 a 9 a 2 a 4 a], b I = [b 0 0 b]$
$A^{2} + a A + b I = [22 + 2 a + b 9 a + 54 12 + 2 a 34 + 4 a + b] = O$
$\Rightarrow a = - 6$
$\Rightarrow 22 + 2 a + b = 0 \Rightarrow b = - 10$
$\Rightarrow a + 2 b = - 26$

Q. If A=[29​24​] and A2+aA+bI=O , then a+2b is equal to (where, I is an identity matrix and O is a null matrix of order 2 respectively)

27

−26

24

12