Question

Let \[ M = \begin{pmatrix} 6 & 2 & -6 & 8 \\ 5 & 3 & -9 & 8 \\ 3 & 1 & -2 & 4 \end{pmatrix} \] Consider the system \( S \) of linear equations given by:
\[ 6x_1 + 2x_2 - 6x_3 + 8x_4 = 8 \] \[ 5x_1 + 3x_2 - 9x_3 + 8x_4 = 16 \] \[ 3x_1 + x_2 - 2x_3 + 4x_4 = 32 \] where \( x_1, x_2, x_3, x_4 \) are unknowns.
Then, which one of the following is TRUE?

• A. The rank of \( M \) is 3, and the system \( S \) has a solution
• B. The rank of \( M \) is 3, and the system \( S \) does NOT have a solution
• C. The rank of \( M \) is 2, and the system \( S \) has a solution
• D. The rank of \( M \) is 2, and the system \( S \) does NOT have a solution

Hint:

Use Gaussian elimination to determine the rank of a matrix. If the rank of the augmented matrix equals the rank of the coefficient matrix, the system has a solution.

Answer 1

The Correct Option is A

Step 1: Form the augmented matrix.
The augmented matrix for the system is: \[ \left( \begin{array}{cccc|c} 6 & 2 & -6 & 8 & 8 \\ 5 & 3 & -9 & 8 & 16 \\ 3 & 1 & -2 & 4 & 32 \end{array} \right) \] Step 2: Apply Gaussian elimination.
First, divide the first row by 6: \[ \left( \begin{array}{cccc|c} 1 & \tfrac{1}{3} & -1 & \tfrac{4}{3} & \tfrac{4}{3} \\ 5 & 3 & -9 & 8 & 16 \\ 3 & 1 & -2 & 4 & 32 \end{array} \right) \] Now, subtract 5 times the first row from the second row and subtract 3 times the first row from the third row: \[ \left( \begin{array}{cccc|c} 1 & \tfrac{1}{3} & -1 & \tfrac{4}{3} & \tfrac{4}{3} \\ 0 & \tfrac{4}{3} & -4 & \tfrac{20}{3} & \tfrac{40}{3} \\ 0 & 0 & 1 & 0 & 24 \end{array} \right) \] Step 3: Final rank determination.
From the row-reduced matrix, the rank of \( M \) is 3, which means the system has a solution.

Final Answer:
\[ \boxed{\text{The rank of } M \text{ is 3, and the system } S \text{ has a solution.}} \]