GATE IN 2023 | Question: 53

Question

The rank of the matrix $A$ given below is one. The ratio $\frac{\alpha}{\beta}$ is __________ (rounded off to the nearest integer).

$$A=\left[\begin{array}{cc}
1 & 4 \\
-3 & \alpha \\
\beta & 6
\end{array}\right]$$

Answer 1

• If the rank of a matrix is 1, all rows must be linearly dependent (scalar multiples of each other).

• Comparing Row 1 and Row 2: $R_2 = -3 \times R_1$. Therefore, $\alpha = -3 \times 4 = -12$.

• Comparing Row 1 and Row 3: The ratio of second elements is $\frac{6}{4} = 1.5$. Thus, $R_3 = 1.5 \times R_1$.

• This gives $\beta = 1.5 \times 1 = 1.5$.

• Calculate the ratio: $\frac{\alpha}{\beta} = \frac{-12}{1.5} = -8$.