The matrix below represents a system of equations.

[tex]\[ \left[\begin{array}{ccc|c} 2 & 1 & 1 & 4 \\ -3 & 2 & -1 & -8 \\ 1 & -1 & 1 & 5 \end{array}\right] \][/tex]

Which matrix represents the solution to this system of equations?

A. [tex]\[ \left[\begin{array}{lll|c} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 3 \end{array}\right] \][/tex]

B. [tex]\[ \left[\begin{array}{ccc|c} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -4 \\ 0 & 0 & 1 & 3 \end{array}\right] \][/tex]



Answer :

To solve the given system of equations, we need to bring the augmented matrix to its reduced row-echelon form (RREF). The original system of equations is represented by the matrix:

[tex]\[ \left[\begin{array}{ccc|c} 2 & 1 & 1 & 4 \\ -3 & 2 & -1 & -8 \\ 1 & -1 & 1 & 5 \end{array}\right] \][/tex]

Following the step-by-step process of Gaussian elimination, the solution to the system can be represented as:

[tex]\[ \left[\begin{array}{ccc|c} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 3 \end{array}\right] \][/tex]

This tells us that:
- The first variable [tex]\( x_1 \)[/tex] is equal to 1,
- The second variable [tex]\( x_2 \)[/tex] is equal to -1,
- The third variable [tex]\( x_3 \)[/tex] is equal to 3.

Therefore, the correct matrix representing the solution to this system of equations is:

[tex]\[ \left[\begin{array}{ccc|c} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 3 \end{array}\right] \][/tex]

The other given matrix:

[tex]\[ \left[\begin{array}{ccc|c} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -4 \\ 0 & 0 & 1 & 3 \end{array}\right] \][/tex]

does not represent the solutions to the system, as the values in the last column (the constants) do not match the solution we derived.