Let S = A = bmatrix a & b c & d bmatrix : a, b, c, d 0, 1, 2, 3, 4 and A^2 - 4A + 3I = 0 be a set of 2 2 matrices. Then the number of matrices in S , for which the sum of the diagonal elements is equal to 4 , is:

Step-by-step solution

The characteristic equation of a 2 2 matrix A = bmatrix a & b c & d bmatrix is given by: A^2 - Tr (A)A + (A)I = 0 We are given that the sum of the diagonal elements is 4 , so Tr (A) = a + d = 4 . Substituting this into the characteristic equation, we get: A^2 - 4A + (ad - bc)I = 0 We are also given that the matrix satisfies: A^2 - 4A + 3I = 0 Comparing the two equations, we must have: ad - bc = 3 We need to find the number of quadruples (a, b, c, d) with elements from the set 0, 1, 2, 3, 4 such that a + d = 4 and a