91Ó°ÊÓ

An augmented matrix is a way to represent a system of linear equations. Including the coefficients of the variables and the constants. The given system\n\[\begin{align*} 3a - b - 4c &= 3, \ 2a - b + 2c &= -8, \ a + 2b - 3c &= 9 \end{align*}\n can be represented as the augmented matrix\n\[\begin{bmatrix} 3 & -1 & -4 & | & 3 \ 2 & -1 & 2 & | & -8 \ 1 & 2 & -3 & | & 9 \end{bmatrix}\n\]

We can use Gauss-Jordan elimination. Row operations are performed until the left side of the augmented matrix is in row-reduced echelon form. This involves getting a 1 in the first row first column by switching the first and third row:\n\[\begin{bmatrix} 1 & 2 & -3 & | & 9 \ 2 & -1 & 2 & | & -8 \ 3 & -1 & -4 & | & 3 \end{bmatrix}\n\]\nProceed with the elimination by subtracting appropriate multiples of the first row from the others to get zeros under the leading 1, then get leading 1 on second row second column:\n\[\begin{bmatrix} 1 & 2 & -3 & | & 9 \ 0 & -5 & 8 & | & -26 \ 0 & -7 & 2 & | & -24 \end{bmatrix}\n\]\nSubtract 1.4 times the second row from the third to get zero in third row, second column:\n\[\begin{bmatrix} 1 & 0 & -5 & | & 6 \ 0 & 1 & -8/5 & | & 26/5 \ 0 & 0 & 2 & | & 0 \end{bmatrix}\n\]\nFinally, divide the last row by 2 for simplicity:\n\[\begin{bmatrix} 1 & 0 & -5 & | & 6 \ 0 & 1 & -8/5 & | & 26/5 \ 0 & 0 & 1 & | & 0 \end{bmatrix}\n\]

The simplified matrix represents the system of equations: \n\[a - 5c = 6, \n b - 8/5 c = 26/5, c = 0.\]

From the last equation, it is clear that \(c = 0\). Substituting \(c = 0\) in the first and second equations, we can solve for \(a\) and \(b\) to get \(a = 6\) and \(b = 26/5\), respectively.