Let be a set of 2 x 2 matrices. Then the number of matrices in S, for which the sum of the diagonal elements is equal to 4, is : [2026]

Question

Let $S = {A = [\begin{matrix} a & b \\ c & d \end{matrix}] : a, b, c, d \in {0, 1, 2, 3, 4} and A^{2} - 4 A + 3 I = 0}$ be a set of 2 x 2 matrices. Then the number of matrices in S, for which the sum of the diagonal elements is equal to 4, is : [2026]

Smart Achievers · Accepted Answer

(4)

We have, Tr = (A) = 4

$\Rightarrow$ a + d = 4 ... (i)

and |A| = 3

[ $∵ A^{2} - 4 A + 3 I = 0$ is characteristic equation of matrix A]

ad –bc = 0

Case 1: If a = 0, d = 4, then bc = –3 (Reject)

Case 2: If a = 1, d = 4, then 3 – bc = 3

$\Rightarrow$ bc = 0

If b = 0, then 'c' has 5 choices and if c = 0, 'b' has 4 choices, so total 9 cases.

Case 3: If a = 2, d = 2, then 4 – bc = 3

$\Rightarrow$ bc = 1 $\Rightarrow$ b = 1, c = 1 (only 1 case)

Case 4: If a = 3, d = 1, then 3 – bc = 3

$\Rightarrow$ bc = 0 $\Rightarrow$ 9 cases

Case 5: If a = 4, d = 0, then bc = 3 (Reject)

$∴$ Total 19 cases.

Solution

Q.
Let $S = {A = [\begin{matrix} a & b \\ c & d \end{matrix}] : a, b, c, d \in {0, 1, 2, 3, 4} and A^{2} - 4 A + 3 I = 0}$ be a set of 2 x 2 matrices. Then the number of matrices in S, for which the sum of the diagonal elements is equal to 4, is : [2026]

1 20

2 17

3 21

4 19

Want Us to Email you About Special Offers & Updates?