SSC IMD SA Mock 03 – Question 141

Find the rank of the given matrix.
\(\begin{bmatrix} 1 & 5 & 6 \\ 2 & 3 & 4 \\ -1 & 2 & 2 \end{bmatrix}\)

• 2
• 6
• 9
• 4

To find the rank of the matrix \(A = \begin{bmatrix} 1 & 5 & 6 \\ 2 & 3 & 4 \\ -1 & 2 & 2 \end{bmatrix}\):

Step 1: Compute the determinant of \(A\):
\[ \det(A) = 1(3 \times 2 – 4 \times 2) – 5(2 \times 2 – 4 \times (-1)) + 6(2 \times 2 – 3 \times (-1)) \]
\[ \det(A) = 1(6 – 8) – 5(4 + 4) + 6(4 + 3) = 1(-2) – 5(8) + 6(7) = -2 – 40 + 42 = 0 \]
Since \(\det(A) = 0\), the rank of \(A\) is strictly less than 3.

Step 2: Check for a non-zero \(2 \times 2\) minor:
Consider the top-left \(2 \times 2\) submatrix:
\[ \begin{vmatrix} 1 & 5 \\ 2 & 3 \end{vmatrix} = (1 \times 3) – (5 \times 2) = 3 – 10 = -7 \neq 0 \]
Since there exists at least one non-zero \(2 \times 2\) minor, the rank of the matrix is 2.
Therefore, the correct answer is option a).