Question

Let A = \(\begin{bmatrix} 1 & 2 & 0 & -1 2 & 6 & -3 & -3 3 & 10 & -6 & -5 \end{bmatrix}\) , then rank of matrix A is :

• A. 0
• B. 2
• C. 4
• D. 3

Hint:

Always look for proportional rows to save time:
Notice that if we subtract Row 1 from Row 3, we get:
\[ [3, 10, -6, -5] - [1, 2, 0, -1] = [2, 8, -6, -4] \]
This is not directly Row 2, but:
\[ R_3 = R_1 + 2 R_2 \]
Verify: \(3 = 1 + 2(1)\), \(10 = 2 + 2(4)\)? Wait: \(1 + 2(2) = 3\), \(2 + 2(6)/ \dots\)
Since \(R_3 = R_1 + 2 R_2 - \dots\) is a linear combination, the rows are linearly dependent, meaning the rank must be less than 3.
Since the rows are not all multiples of each other, the rank must be exactly 2.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The rank of a matrix is the maximum number of linearly independent row vectors (or column vectors) in the matrix.
To find the rank, we perform elementary row operations to reduce the matrix to row-echelon form.
The number of non-zero rows in the row-echelon form is equal to the rank of the matrix.

Step 2: Detailed Explanation:
Let the given matrix be:
\[ A = \begin{bmatrix} 1 & 2 & 0 & -1 2 & 6 & -3 & -3 3 & 10 & -6 & -5 \end{bmatrix} \]
We will perform elementary row operations to reduce it:
Let \(R_1, R_2, R_3\) denote the rows of the matrix.
-

Step 1: Eliminate the first entry of the second row \(R_2\).
Perform the operation: \(R_2 \to R_2 - 2R_1\)
\[ R_2 - 2R_1 = [2, 6, -3, -3] - 2[1, 2, 0, -1] \]
\[ = [2, 6, -3, -3] - [2, 4, 0, -2] = [0, 2, -3, -1] \]
- Eliminate the first entry of the third row \(R_3\).
Perform the operation: \(R_3 \to R_3 - 3R_1\)
\[ R_3 - 3R_1 = [3, 10, -6, -5] - 3[1, 2, 0, -1] \]
\[ = [3, 10, -6, -5] - [3, 6, 0, -3] = [0, 4, -6, -2] \]
The matrix now becomes:
\[ A' = \begin{bmatrix} 1 & 2 & 0 & -1 0 & 2 & -3 & -1 0 & 4 & -6 & -2 \end{bmatrix} \]
-

Step 2: Eliminate the second entry of the third row \(R_3\).
Observe that the new third row \([0, 4, -6, -2]\) is exactly twice the new second row \([0, 2, -3, -1]\).
Perform the operation: \(R_3 \to R_3 - 2R_2\)
\[ R_3 - 2R_2 = [0, 4, -6, -2] - 2[0, 2, -3, -1] \]
\[ = [0, 4, -6, -2] - [0, 4, -6, -2] = [0, 0, 0, 0] \]
The matrix in row-echelon form is:
\[ \begin{bmatrix} 1 & 2 & 0 & -1 0 & 2 & -3 & -1 0 & 0 & 0 & 0 \end{bmatrix} \]
The reduced matrix has exactly 2 non-zero rows.
Therefore, the rank of matrix \(A\) is 2.

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