JEE Mains · Maths · STD 12 - 3 and 4 . metrices and determinant
Let \(S = \left\{A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} : a, b, c, d \in \{0, 1, 2, 3, 4\} \text{ and } A^2 - 4A + 3I = 0\right\}\) be a set of \(2 \times 2\) matrixes. Then the number of matrixes in \(S\), for which the sum of the diagonal elements is equal to \(4\), is:
- A \(20\)
- B \(17\)
- C \(21\)
- D \(19\)
Answer & Solution
Correct Answer
(D) \(19\)
Step-by-step Solution
Detailed explanation
The characteristic equation of a \(2 \times 2\) matrix \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\) is given by: \(A^2 - \text{Tr}(A)A + \det(A)I = 0\) We are given that the sum of the diagonal elements is \(4\), so \(\text{Tr}(A) = a + d = 4\). Substituting this into…
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