Question

The rank of the matrix \[ A= \begin{bmatrix} 2 & 1 & 2 \\ 1 & 0 & 1 \\ 4 & 1 & 4 \end{bmatrix} \] is

• A. \(0\)
• B. \(1\)
• C. \(2\)
• D. \(3\)

Hint:

For a \(3\times 3\) matrix, if the determinant is zero, the rank is less than \(3\). Then check whether any \(2\times 2\) minor is non-zero. If yes, the rank is \(2\).

Answer 1

The Correct Option is C

Step 1: Check if the determinant is zero.
Notice that row 3 = \(2\times\) row 1 (since \((4,1,4) = 2\times(2,1,2)\) — wait, \(2\times1=2\neq1\)). Actually column 3 = column 1 for all rows: col1 = (2,1,4) and col3 = (2,1,4). So two columns are identical, giving \(\det(A) = 0\).

Step 2: Find a non-zero 2x2 minor.
Take the top-left submatrix \(\begin{bmatrix}2&1\\1&0\end{bmatrix}\); its determinant is \(0-1=-1\neq 0\). So rank is exactly 2.
\[\boxed{\text{rank}(A) = 2}\]