GATE CS 2002 Q1.1 — Linear Algebra: Matrices, Determinants, System of Linear Equations

The rank of a matrix is defined as the maximum number of linearly independent row (or column) vectors in the matrix.

Let the given matrix be:

$$A = \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}$$

Let us analyze the row vectors of matrix $A$:

• $R_1 = [1, 1]$ (a non-zero row)
• $R_2 = [0, 0]$ (a zero row)

Since $R_2$ is a zero vector, it is linearly dependent. The only linearly independent row is $R_1$.

Therefore, the number of linearly independent rows is $1$, which means the rank of the matrix is $1$.

Thus, the correct option is C.