Answer to Question #348294 in Linear Algebra for Shahzadi

Question #348294

2X1+2X3-x3+x5=6,-X1-x2+2X3-3X4+X5=0,X3+X4+X5=0,X3+X5+X5=0 SOLVE IT BY USING GUASS JORDON METHOD

1
Expert's answer
2022-06-14T12:12:11-0400

As far as we have 4 equations and 5 variables the system has an infinite number of solutions. But let's find one of them.


Let's create an coefficient's matrix:


"\\begin{pmatrix}\n 2 &0&1&0&1&6 \\\\\n -1 &-1&2&3&1&0 \\\\\n 0&0&1&1&1&0 \\\\\n 0&0&1&0&2&0\n\\end{pmatrix}"


after that according to an Gauss algorithm, we need to made an 1 in upper left corner, let's make it by adding second row to the first one (than I'll write it as I = I + II, witch means, that the first row is the sum of first and second etc)


We'll get next matrix:

"\\begin{pmatrix}\n 1 &-1&3&3&2&6 \\\\\n -1 &-1&2&3&1&0 \\\\\n 0&0&1&1&1&0 \\\\\\\n\n 0&0&1&0&2&0\n\\end{pmatrix}"


Then we need to have 0's in first column. II = II + I

"\\begin{pmatrix}\n 1 &-1&3&3&2&6 \\\\\n 0&-2&5&6&3&6 \\\\\n 0&0&1&1&1&0 \\\\\\\n\n 0&0&1&0&2&0\n\\end{pmatrix}"


Then to have 1 in second column and second row let's make next transformations:


II = II + III

"\\begin{pmatrix}\n 1 &-1&3&3&2&6 \\\\\n 0&-2&6&7&4&6 \\\\\n 0&0&1&1&1&0 \\\\\\\n\n 0&0&1&0&2&0\n\\end{pmatrix}"


II = II / (-2)

"\\begin{pmatrix}\n 1 &-1&3&3&2&6 \\\\\n 0&1&-3&-3.5&-2&-3\n \\\\\n 0&0&1&1&1&0 \\\\\\\n\n 0&0&1&0&2&0\n\\end{pmatrix}"


To make a 0 in a first row second column we'll make I = I + II

"\\begin{pmatrix}\n 1 &0&0&-0.5&0&3 \\\\\n 0&1&-3&-3.5&-2&-3\n \\\\\n 0&0&1&1&1&0 \\\\\\\n\n 0&0&1&0&2&0\n\\end{pmatrix}"


Replace 3-rd row with 4-th:

"\\begin{pmatrix}\n 1 &0&0&-0.5&0&3 \\\\\n 0&1&-3&-3.5&-2&-3 \\\\\n 0&0&1&0&2&0 \\\\\n 0&0&1&1&1&0 \n\\end{pmatrix}"


II = II + 3*III

IV = IV - III

"\\begin{pmatrix}\n 1 &0&0&-0.5&0&3 \\\\\n 0&1&0&-3.5&4&-3 \\\\\n 0&0&1&0&2&0 \\\\\n 0&0&0&1&-1&0 \n\\end{pmatrix}"


I = I + 0.5*IV

II = II + 3.5*IV

"\\begin{pmatrix}\n 1 &0&0&0&-0.5&3 \\\\\n 0&1&0&0&0.5&-3 \\\\\n 0&0&1&0&2&0 \\\\\n 0&0&0&1&-1&0 \n\\end{pmatrix}"


Let x5 = 2, then:

x1 - 0.5*x5 = 3 ==> x1 = 4

x2 + 0.5*x5 = -3 ==> x2 = -4

x3 + 2*x5 = 0 ==> x3 = -4

x4 - x5 = 0 ==> x4 = 2


One of the solutions is {4, -4, -4, 2, 2}



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