Question

Rank of the matrix $A = \begin{pmatrix} 2 & 3 & -1 \\ -8 & -7 & 6 \\ 6 & -1 & -7 \end{pmatrix}$ is

• A. 2
• B. 1
• C. 3
• D. 0

Hint:

To find the rank of a $3\times3$ matrix, always compute the overall determinant first. If it is non-zero, the rank is 3; if it is zero, search for at least one $2\times2$ submatrix with a non-zero determinant to establish a rank of 2.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The rank of a matrix is the maximum number of linearly independent row or column vectors in the matrix, which is also equal to the order of the largest non-zero minor.

Step 2: Detailed Explanation:
Let us calculate the determinant of the $3 \times 3$ matrix $A$:
\[ |A| = 2 \cdot ((-7) \cdot (-7) - 6 \cdot (-1)) - 3 \cdot ((-8) \cdot (-7) - 6 \cdot 6) + (-1) \cdot ((-8) \cdot (-1) - (-7) \cdot 6) \]
\[ |A| = 2 \cdot (49 + 1) - 3 \cdot (56 - 36) - 1 \cdot (8 + 42) \]
Wait, let us re-verify the multiplication:
For the first term:
\[ (-7) \cdot (-7) - 6 \cdot (-1) = 49 + 6 = 55 \]
Substitute the correct values back:
\[ |A| = 2 \cdot (55) - 3 \cdot (20) - 1 \cdot (50) \]
\[ |A| = 110 - 60 - 50 = 0 \]
Since the determinant of the $3 \times 3$ matrix is zero, the rank of $A$ must be less than 3.
Now let us check if there is any non-zero $2 \times 2$ minor:
Consider the top-left submatrix:
\[ M = \begin{pmatrix} 2 & 3 -8 & -7 \end{pmatrix} \]
Calculate its determinant:
\[ |M| = 2 \cdot (-7) - 3 \cdot (-8) = -14 + 24 = 10 \]
Since $10 \neq 0$, we have found a non-zero minor of order 2.
Therefore, the rank of the matrix $A$ is 2.

Step 3: Final Answer
The correct option is (A).