GATE CS 1998 Q2.1 — Linear Algebra: Matrices, Determinants, System of Linear Equations

To find the rank of the matrix $A$, we perform elementary row operations:

$$A = \begin{bmatrix} 1 & 4 & 8 & 7 \\ 0 & 0 & 3 & 0 \\ 4 & 2 & 3 & 1 \\ 3 & 12 & 24 & 21 \end{bmatrix}$$

Observe that the fourth row $R_4$ is a scalar multiple of the first row $R_1$, specifically $R_4 = 3R_1$. Applying the row operation $R_4 \to R_4 - 3R_1$ gives:

$$\begin{bmatrix} 1 & 4 & 8 & 7 \\ 0 & 0 & 3 & 0 \\ 4 & 2 & 3 & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$

To determine if the remaining three rows are linearly independent, we compute the determinant of the $3 \times 3$ submatrix formed by the first three columns:

$$M = \begin{bmatrix} 1 & 4 & 8 \\ 0 & 0 & 3 \\ 4 & 2 & 3 \end{bmatrix}$$

Expanding along the second row:

$$\det(M) = -3 \begin{vmatrix} 1 & 4 \\ 4 & 2 \end{vmatrix} = -3(2 - 16) = 42 \neq 0$$

Since there exists a non-zero minor of order 3, the rank of the matrix is 3.