GATE CS 2026 Set 2 Q21 — Linear Algebra: Matrices, Determinants, System of Linear Equations

For a system of two linear equations in two variables, $a_1 x + b_1 y = c_1$ and $a_2 x + b_2 y = c_2$, to have infinitely many (multiple) solutions, the equations must represent the same line. This requires the coefficients and constants to be proportional:

Given the system of equations:

Applying the condition for multiple solutions, we get:

From the first equality, we have $a^2 = 16$, which implies $a \neq 0$.

From the second equality, we have:

Multiplying both sides by $a$ gives:

Thus, the product of $a$ and $b$ is 24.