If A= 1 -3 2 k and A^2-4A+10I=A, then k is equal to

Correct answer: (C) 4. A^2-4A+10I=A ⇒ 1 -3 2 k 1 -3 2 k -4 1 -3 2 k +10 1 0 0 1 = 1 -3 2 k ⇒ −5 −3−3k 2+2k −6+k^2 − 4 −12 8 4k + 10 0 0 10 = 1 −3 2 k ⇒ 1 9-3k -6+2k 4+k^2-4k = 1 -3 2 k ⇒9-3k=-3,-6+2k=2 ...(i) and 4+k^2-4k=k ⇒k^2 −5k+4=0⇒k=4, 1 But, k=1 does not satisfy the Eq (i).

Q38 / 75 · NTA Abhyas (JEE Main) 2020 · Mathematics · Matrices

If $A = [\begin{matrix} 1 & - 3 \\ 2 & k \end{matrix}]$ and $A^{2} - 4 A + 10 I = A$ , then $k$ is equal to

A
$0$
B
$- 4$
Correct answer
C
$4$
D
$1$ or $4$

Solution

A^{2} - 4 A + 10 I = A

\Rightarrow [\begin{matrix} 1 - 3 \\ 2 k \end{matrix}] [\begin{matrix} 1 - 3 \\ 2 k \end{matrix}] - 4 [\begin{matrix} 1 - 3 \\ 2 k \end{matrix}] + 10 [\begin{matrix} 1 0 \\ 0 1 \end{matrix}] = [\begin{matrix} 1 - 3 \\ 2 k \end{matrix}]

\Rightarrow [\begin{matrix} - 5 & - 3 - 3 k \\ 2 + 2 k & - 6 + k^{2} \end{matrix}] - [\begin{matrix} 4 & - 12 \\ 8 & 4 k \end{matrix}] + [\begin{matrix} 10 & 0 \\ 0 & 10 \end{matrix}] = [\begin{matrix} 1 & - 3 \\ 2 & k \end{matrix}]

\Rightarrow [\begin{matrix} 1 9 - 3 k \\ - 6 + 2 k 4 + k^{2} - 4 k \end{matrix}] = [\begin{matrix} 1 - 3 \\ 2 k \end{matrix}]

\Rightarrow 9 - 3 k = - 3, - 6 + 2 k = 2

...(i)

and

4 + k^{2} - 4 k = k

\Rightarrow k^{2} - 5 k + 4 = 0 \Rightarrow k = 4, 1

But,

k = 1

does not satisfy the Eq (i).

Hence, the correct answer is option: 'C'.

If A=1-32 k and A2-4A+10I=A, then k is equal to

Solution

More Matrices Questions

If $A = [\begin{matrix} 1 & - 3 \\ 2 & k \end{matrix}]$ and $A^{2} - 4 A + 10 I = A$ , then $k$ is equal to