Question

The rank of the following matrix is:
\[\begin{pmatrix} 1 & 5 & 1 \\ 2 & 1 & 1 \\ 5 & 6 & 2 \end{pmatrix}\]

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

For a \(3\times 3\) matrix, evaluate the determinant. If it is non-zero the rank is 3.

Answer 1

The Correct Option is C

Step 1: For a \(3\times 3\) matrix, the rank is 3 if and only if the determinant is non-zero. Compute the determinant by expanding along the first row:
\[\det = 1\,(1\cdot 2 - 1\cdot 6) - 5\,(2\cdot 2 - 1\cdot 5) + 1\,(2\cdot 6 - 1\cdot 5)\]
Step 2: Evaluate each minor:
\[= 1\,(2 - 6) - 5\,(4 - 5) + 1\,(12 - 5)\]
\[= 1\,(-4) - 5\,(-1) + 1\,(7) = -4 + 5 + 7 = 8\]
Step 3: Since \(\det = 8 \neq 0\), all three rows (and columns) are linearly independent, so the matrix has full rank.
\[\boxed{\text{rank} = 3}\]