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Chemical Engineering Numerical Methods - Systems for Linear Equations Experiments

System of Linear Equations - Gaussian Elimination

\begin{bmatrix} -127.7552 & 34.1582 & -19.229 \\ -30.7677 & -34.5364 & -40.5769 \\ -49.7605 & 32.148 & -31.3791 \end{bmatrix} \cdot X = \begin{bmatrix}-63.4307 \\ -88.3105 \\ 26.7046 \end{bmatrix}

Form the upper triangular matrix

a:

\begin{bmatrix} -120.552 & 40.6582 & -13.829 \\ 0 & -40.728 & -32.9439 \\ 0 & 0 & -44.7819 \end{bmatrix} \cdot X = \begin{bmatrix} -67.4707 \\ -72.0312 \\ 22.8282 \end{bmatrix}

Explanation

Explanation

b:

\begin{bmatrix} -127.7552 & 34.1582 & -19.229 \\ 0 & -42.7628 & -35.9459 \\ 0 & 0 & -39.7289 \end{bmatrix} \cdot X = \begin{bmatrix} -63.4307 \\ -73.0342 \\ 19.2282 \end{bmatrix}

Explanation

Explanation

c:

\begin{bmatrix} -122.7152 & 41.1158 & -18.259 \\ 0 & -47.7128 & -39.9419 \\ 0 & 0 & -31.2859 \end{bmatrix} \cdot X = \begin{bmatrix} -65.2307 \\ -71.0442 \\ 11.2782 \end{bmatrix}

Explanation

Explanation

d:

\begin{bmatrix} -131.5352 & 42.6582 & -12.129 \\ 0 & -32.2218 & -41.4559 \\ 0 & 0 & -32.7189 \end{bmatrix} \cdot X = \begin{bmatrix} -59.407 \\ -76.0152 \\ 23.2852 \end{bmatrix}

Explanation

Explanation

Perform back-substitution and find X.

\begin{bmatrix}191.7248 & -31.599 & 38.4242 \\ -41.2838 & -26.6322 & 49.8417 \\ 37.6895 & 33.558 & 10.8116 \end{bmatrix} \cdot X = \begin{bmatrix} 74.8596 \\ -41.1049 \\ -48.1611\end{bmatrix}

a:

\begin{bmatrix} 0.1276 \\ -4.3732 \\ -7.7245\end{bmatrix}

Explanation

Explanation

b:

\begin{bmatrix} 0.2316 \\ -8.8132 \\ -8.4125\end{bmatrix}

Explanation

Explanation

c:

\begin{bmatrix} 0.1636 \\ -6.7932 \\ -4.4545\end{bmatrix}

Explanation

Explanation

d:

\begin{bmatrix} 0.2156 \\ -5.7232 \\ -3.5438\end{bmatrix}

Explanation

Explanation