A.kaw Matrix Algebra Preliminary study note 6. Gaussian Elimination

"Matrix Algebra Preliminary" (Introduction to Matrix ALGEBRA) course by Prof. A.k.kaw (University of South Florida) is designed and taught.
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Summary

• Gaussian elimination consists of the steps:
  • Forward Elimination of unknowns In this step, the unknown are eliminated in each equation starting with the first equation. This is the equations is reduced to one equation and one unknown in each equation.
  • Back Substitution In this step, starting from the last equation, each of the unknowns is found.
• More about determinant
  • Let $[a]$ is A $n \times n$ matrix. Then if $[b]$ is a $n \times n$ matrix the results from adding or subtracting a multiple of one row (column) to another RO W (column), then $\det (A) = \det (B) $.
  • Let $[a]$ are A $n \times n$ matrix that's upper triangular, lower triangular or diagonal, then $$\det (a) = A_{11}\times A_ {22}\times\cdots\times A_{nn} = \prod_{i=1}^{n}a_{ii}$$ This implies if we apply the forward elimination steps of Gau Ssian elimination method, the determinant of the matrix stays the same according to the previous result. Then since at the end of the forward elimination steps, the resulting matrix is upper triangular, the determinant would be Given by the above result.

Selected problems

1. Using Gaussian elimination to solve $$\begin{cases}4x_1+x_2-x_3=-2\\ 5x_1+x_2+2x_3=4\\ 6x_1+x_2+x_3=6\end{cases}$$

Solution:

Forward Elimination: $$\begin{bmatrix}4& 1& -1& -2\\ 5& 1& 2& 4\\ 6& 1& 1& 6\end{bmatr Ix}\rightarrow \begin{cases} r_2-{5\over4}r_1\\ r_3-{3\over2}r_1\end{cases}\begin{bmatrix}4& 1& -1& -2\\ 0&-{1\over4}& {13\over4}& {13\over2}\\ 0&-{1\over2}& {5\over2}& 9\end{bmatrix}$$ $$\ RightArrow r_3-2r_2\begin{bmatrix}4& 1& -1& -2\\ 0&-{1\over4}& {13\over4}& {13\over2}\\ 0& 0& -4& -4\end{bmatrix}$$ back substitution: $$\begin{cases}-4x_3=-4\\-{1\over4}x_2+{13\over4}x_3={13\over2} \ 4x_1+x_2-x_3=-2\end{cases} \rightarrow \begin{cases}x_3=1\\-{1\over4}x_2+{13\over4}={13\over2}\\ 4x_1+x_2-1=-2\ end{cases}$$ $$\rightarrow \begin{cases}x_3=1\\ x_2 = -13\\ 4x_1-13=-1 \end{cases}\rightarrow \begin{cases}x_1 = 3\\ x_2= -13\\ x_3=1 \end{cases}$$

2. Find the determinant of $$[a] = \begin{bmatrix}25& 5& 1\\ 64& 8& 1\\ 144& 12& 1\end{bmatrix}$$

Solution:

Forward elimination $$[a] = \begin{bmatrix}25& 5& 1\\ 64& 8& 1\\ 144& 12& 1\end{bmatrix}\Rightarro W\begin{cases}r_2-{64\over25}r_1\\ r_3-{144\over25}r_1\end{cases} \begin{bmatrix}25& 5& 1\\ 0&-{24\ over5}&-{39\over25}\\ 0&-{84\over5}&-{119\over25} \end{bmatrix}$$ $$\rightarrow R_3-{7\over2}R_2 \begin {bmatrix}25& 5& 1\\ 0&-{24\over5}&-{39\over25}\\ 0& 0 & {7\over10} \end{bmatrix}$$ Per triangular matrix and its determinant are the product of the diagonal elements $$\det (A) = 25\times (-{24\over5}) \times{ 7\over10}=-84 $$

3. Find the determinant of $$[a] = \begin{bmatrix}10& -7& 0\\ -3& 2.099& 6\\ 5& -1& 5\end{bmatrix} $$

Solution:

Forward elimination $$[a] = \begin{bmatrix}10& -7& 0\\ -3& 2.099& 6\\ 5& -1& 5 \end{bmatrix}\Right arrow\begin{cases}r_2 + {3\over 10}r_1\\ r_3-{1\over2}r_1\end{cases} \begin{bmatrix}10& -7& 0\\ 0&-{1\ over1000}& 6\\ 0& {5\over2}& 5 \end{bmatrix}$$ $$\rightarrow r_3+2500r_2 \begin{bmatrix}10& -7& 0\\ 0 &-{1\over1000}& 6\\ 0& 0 & 15005 \end{bmatrix}$$ This is a upper triangular matrix and its determinant I s the product of the diagonal elements $$\det (A) = ten \times (-{1\over1000}) \times15005=-150.05$$

4. Using Gaussian elimination to solve $$\begin{cases}3x_1-x_2-5x_3 = 9\\ x_2-10x_3=0\\ -2x_1+x_2=-6\end{cases}$$

Solution:

Forward Elimination: $$\begin{bmatrix}3& -1& -5& 9\\ 0& 1& -10& 0\\ -2& 1& 0& -6\end{b Matrix}\rightarrow r_3+{2\over3}r_1 \begin{bmatrix}3& -1& -5& 9\\ 0& 1& -10& 0\\ 0& {1\over3} &-{10\over3}& 0\end{bmatrix}$$ $$\rightarrow r_3-{1\over3}r_2 \begin{bmatrix}3& -1& -5& 9\\ 0& 1& -10& 0\\ 0& 0 & 0 & 0\end{bmatrix}$$ back substitution: $$\begin{cases}x_2-10x_3=0\\ 3x_1-x_2-5x_3= 9\end{cases} \rightarrow \begin{cases}x_2 = 10x_3\\ 3x_1-15x_3 = 9\end{cases} \rightarrow \begin{cases}x_1 = 5x_3+3\\ x_2 = 10x_3\end{cases}$$ where $x _3$ is arbitrary.

A.kaw Matrix Algebra Preliminary study note 6. Gaussian Elimination