Matrices And Determinants

Step 1: Condition for a Singular Matrix

A matrix is singular if its determinant is zero. Thus, we need to evaluate:

Step 2: Expanding the Determinant

Expanding along the first row:

Step 3: Evaluating the Minors

Calculating the determinant of the 2×2 matrices:

Substituting back:

Step 4: Calculating 2α + 3β + 4γ

Given that the required expression evaluates to 4, the final answer is:

4