GATE 2017 EC (EC2) – Question 11
The rank of the matrix
$$\begin{bmatrix}1&-1&0&0&0\\0&0&1&-1&0\\0&1&-1&0&0\\-1&0&0&0&1\\0&0&0&1&-1\end{bmatrix}$$
is ________.
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Show answer and explanation
Correct answer: 4 to 4
Explanation
Solve $Ax=0$. The rows successively require $x_1=x_2$, $x_3=x_4$, $x_2=x_3$, $x_5=x_1$ and $x_4=x_5$. Hence all five coordinates must be equal.
The nullspace is exactly the one-dimensional span of $(1,1,1,1,1)^T$. Rank–nullity therefore gives $\operatorname{rank}(A)=5-1=$ **4**.