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:

Notice that the third row of the original matrix is a linear combination of the first two rows:
\[ R_3 = R_1 + R_2 \] Specifically, \([1, 2, 0, -1] + [2, 6, -3, -3] = [3, 10, -6, -5] = R_3\).
Since the third row is linearly dependent, the rank must be less than 3. Since the first two rows are clearly not multiples of each other, the rank must be exactly 2.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The rank of a matrix is the maximum number of linearly independent row vectors (or column vectors) in the matrix.
To find the rank of a matrix, we can reduce it to row echelon form using elementary row operations.
Key Formula or Approach:
Perform Gaussian elimination (elementary row operations) on matrix \(A\):
- Row replacement: \(R_i \to R_i - c R_j\)
The number of non-zero rows in the resulting echelon form will be the rank of the matrix.

Step 2: Detailed Explanation:

• Write down the original matrix \(A\):
\[ A = \begin{bmatrix} 1 & 2 & 0 & -1 2 & 6 & -3 & -3 3 & 10 & -6 & -5 \end{bmatrix} \]

• We want to create zeros below the leading 1 in the first column.
Perform the row operations:
\[ R_2 \to R_2 - 2R_1 \] \[ R_3 \to R_3 - 3R_1 \]

• Calculating the new rows:
For \(R_2\):
\[ R_2 - 2R_1 = [2, 6, -3, -3] - [2, 4, 0, -2] = [0, 2, -3, -1] \] For \(R_3\):
\[ R_3 - 3R_1 = [3, 10, -6, -5] - [3, 6, 0, -3] = [0, 4, -6, -2] \] Thus, the matrix becomes:
\[ \begin{bmatrix} 1 & 2 & 0 & -1 0 & 2 & -3 & -1 0 & 4 & -6 & -2 \end{bmatrix} \]

• Now, we want to create a zero below the leading entry in the second column.
Perform the row operation:
\[ R_3 \to R_3 - 2R_2 \]

• Calculating the new \(R_3\):
\[ R_3 - 2R_2 = [0, 4, -6, -2] - [0, 4, -6, -2] = [0, 0, 0, 0] \] Thus, the row echelon form of the matrix is:
\[ \begin{bmatrix} 1 & 2 & 0 & -1 0 & 2 & -3 & -1 0 & 0 & 0 & 0 \end{bmatrix} \]

• Count the number of non-zero rows in the echelon form.
There are 2 non-zero rows (Row 1 and Row 2).
Therefore, the rank of the matrix is 2.

Step 3: Final Answer:
The rank of matrix \(A\) is 2.