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 $$2 \times 2$$ matrices. Then the number of matrices in $$S$$, for which the sum of the diagonal elements is equal to 4, is :

For any $$2 \times 2$$ matrix $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$, the characteristic equation is given by:

$$A^2 - \text{tr}(A)A + \det(A)I = 0$$

where $$\text{tr}(A) = a + d$ and $\det(A) = ad - bc$$.

The problem gives us the matrix equation:

$$A^2 - 4A + 3I = 0$$

By comparing the two equations, we find two necessary conditions for $$A$$:

1. Trace: $$\text{tr}(A) = a + d = 4$$
2. Determinant: $$\det(A) = ad - bc = 3$$

We are given $$a, b, c, d \in \{0, 1, 2, 3, 4\}$$. From the trace condition $$a + d = 4$$, the possible pairs $$(a, d)$$ are:

• $$(0, 4)$$
• $$(1, 3)$$
• $$(2, 2)$$
• $$(3, 1)$$
• $$(4, 0)$$

For each pair $$(a, d)$$, we use the determinant condition $$bc = ad - 3$$ to find the number of possible pairs $$(b, c)$$.

Summing the counts from each case:

$$\text{Total number of matrices} = 9 + 1 + 9 = 19$$

The number of matrices in $$S$$ for which the sum of the diagonal elements is equal to $$4$$ is 19.