Answer:
a. The solutions are
![\left[\begin{array}{c}x_1&x_2&x_3\\\end{array}\right]=\begin{pmatrix}\frac{11}{13}\\ \frac{50}{13}\\ -\frac{17}{13}\end{pmatrix}](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Dx_1%26x_2%26x_3%5C%5C%5Cend%7Barray%7D%5Cright%5D%3D%5Cbegin%7Bpmatrix%7D%5Cfrac%7B11%7D%7B13%7D%5C%5C%20%5Cfrac%7B50%7D%7B13%7D%5C%5C%20-%5Cfrac%7B17%7D%7B13%7D%5Cend%7Bpmatrix%7D)
b. The solutions are
![\left[\begin{array}{c}x&y&z\\\end{array}\right]=\begin{pmatrix}\frac{54}{235}\\ \frac{6}{47}\\ \frac{24}{235}\end{pmatrix}](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Dx%26y%26z%5C%5C%5Cend%7Barray%7D%5Cright%5D%3D%5Cbegin%7Bpmatrix%7D%5Cfrac%7B54%7D%7B235%7D%5C%5C%20%5Cfrac%7B6%7D%7B47%7D%5C%5C%20%5Cfrac%7B24%7D%7B235%7D%5Cend%7Bpmatrix%7D)
c. The solutions are
![\left[\begin{array}{c}x_1&x_2&x_3&x_4\\\end{array}\right]=\begin{pmatrix}\frac{22}{9}\\ \frac{164}{9}\\ \frac{139}{9}\\ -\frac{37}{3}\end{pmatrix}](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Dx_1%26x_2%26x_3%26x_4%5C%5C%5Cend%7Barray%7D%5Cright%5D%3D%5Cbegin%7Bpmatrix%7D%5Cfrac%7B22%7D%7B9%7D%5C%5C%20%5Cfrac%7B164%7D%7B9%7D%5C%5C%20%5Cfrac%7B139%7D%7B9%7D%5C%5C%20-%5Cfrac%7B37%7D%7B3%7D%5Cend%7Bpmatrix%7D)
Step-by-step explanation:
Solving a system of linear equations using matrix method, we may define a system of equations with the same number of equations as variables as:
![A\cdot X=B](https://tex.z-dn.net/?f=A%5Ccdot%20X%3DB)
where X is the matrix representing the variables of the system, B is the matrix representing the constants, and A is the coefficient matrix.
Then the solution is this:
![X=A^{-1}B](https://tex.z-dn.net/?f=X%3DA%5E%7B-1%7DB)
a. Given the system:
![3x_1 + 2x_2 + 4x_3 = 5 \\2x_1 + 5x_2 + 3x_3 = 17 \\7x_1 + 2x_2 + 2x_3 = 11](https://tex.z-dn.net/?f=3x_1%20%2B%202x_2%20%2B%204x_3%20%3D%205%20%5C%5C2x_1%20%2B%205x_2%20%2B%203x_3%20%3D%2017%20%5C%5C7x_1%20%2B%202x_2%20%2B%202x_3%20%3D%2011)
The coefficient matrix is:
![A=\left[\begin{array}{ccc}3&2&4\\2&5&3\\7&2&2\end{array}\right]](https://tex.z-dn.net/?f=A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D3%262%264%5C%5C2%265%263%5C%5C7%262%262%5Cend%7Barray%7D%5Cright%5D)
The variable matrix is:
![X=\left[\begin{array}{c}x_1&x_2&x_3\\\end{array}\right]](https://tex.z-dn.net/?f=X%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Dx_1%26x_2%26x_3%5C%5C%5Cend%7Barray%7D%5Cright%5D)
The constant matrix is:
![B=\left[\begin{array}{c}5&17&11\\\end{array}\right]](https://tex.z-dn.net/?f=B%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D5%2617%2611%5C%5C%5Cend%7Barray%7D%5Cright%5D)
First, we need to find the inverse of the A matrix. To find the inverse matrix, augment it with the identity matrix and perform row operations trying to make the identity matrix to the left. Then to the right will be inverse matrix.
So, augment the matrix with identity matrix:
![\left[ \begin{array}{ccc|ccc}3&2&4&1&0&0 \\\\ 2&5&3&0&1&0 \\\\ 7&2&2&0&0&1\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bccc%7Cccc%7D3%262%264%261%260%260%20%5C%5C%5C%5C%202%265%263%260%261%260%20%5C%5C%5C%5C%207%262%262%260%260%261%5Cend%7Barray%7D%5Cright%5D)
This matrix can be transformed by a sequence of elementary row operations to the matrix
![\left[ \begin{array}{ccc|ccc}1&0&0&- \frac{2}{39}&- \frac{2}{39}&\frac{7}{39} \\\\ 0&1&0&- \frac{17}{78}&\frac{11}{39}&\frac{1}{78} \\\\ 0&0&1&\frac{31}{78}&- \frac{4}{39}&- \frac{11}{78}\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bccc%7Cccc%7D1%260%260%26-%20%5Cfrac%7B2%7D%7B39%7D%26-%20%5Cfrac%7B2%7D%7B39%7D%26%5Cfrac%7B7%7D%7B39%7D%20%5C%5C%5C%5C%200%261%260%26-%20%5Cfrac%7B17%7D%7B78%7D%26%5Cfrac%7B11%7D%7B39%7D%26%5Cfrac%7B1%7D%7B78%7D%20%5C%5C%5C%5C%200%260%261%26%5Cfrac%7B31%7D%7B78%7D%26-%20%5Cfrac%7B4%7D%7B39%7D%26-%20%5Cfrac%7B11%7D%7B78%7D%5Cend%7Barray%7D%5Cright%5D)
And the inverse of the A matrix is
![A^{-1}=\left[ \begin{array}{ccc} - \frac{2}{39} & - \frac{2}{39} & \frac{7}{39} \\\\ - \frac{17}{78} & \frac{11}{39} & \frac{1}{78} \\\\ \frac{31}{78} & - \frac{4}{39} & - \frac{11}{78} \end{array} \right]](https://tex.z-dn.net/?f=A%5E%7B-1%7D%3D%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bccc%7D%20-%20%5Cfrac%7B2%7D%7B39%7D%20%26%20-%20%5Cfrac%7B2%7D%7B39%7D%20%26%20%5Cfrac%7B7%7D%7B39%7D%20%5C%5C%5C%5C%20-%20%5Cfrac%7B17%7D%7B78%7D%20%26%20%5Cfrac%7B11%7D%7B39%7D%20%26%20%5Cfrac%7B1%7D%7B78%7D%20%5C%5C%5C%5C%20%5Cfrac%7B31%7D%7B78%7D%20%26%20-%20%5Cfrac%7B4%7D%7B39%7D%20%26%20-%20%5Cfrac%7B11%7D%7B78%7D%20%5Cend%7Barray%7D%20%5Cright%5D)
Next, multiply
by ![B](https://tex.z-dn.net/?f=B)
![X=A^{-1}\cdot B](https://tex.z-dn.net/?f=X%3DA%5E%7B-1%7D%5Ccdot%20B)
![\left[\begin{array}{c}x_1&x_2&x_3\\\end{array}\right]=\left[ \begin{array}{ccc} - \frac{2}{39} & - \frac{2}{39} & \frac{7}{39} \\\\ - \frac{17}{78} & \frac{11}{39} & \frac{1}{78} \\\\ \frac{31}{78} & - \frac{4}{39} & - \frac{11}{78} \end{array} \right] \cdot \left[\begin{array}{c}5&17&11\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Dx_1%26x_2%26x_3%5C%5C%5Cend%7Barray%7D%5Cright%5D%3D%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bccc%7D%20-%20%5Cfrac%7B2%7D%7B39%7D%20%26%20-%20%5Cfrac%7B2%7D%7B39%7D%20%26%20%5Cfrac%7B7%7D%7B39%7D%20%5C%5C%5C%5C%20-%20%5Cfrac%7B17%7D%7B78%7D%20%26%20%5Cfrac%7B11%7D%7B39%7D%20%26%20%5Cfrac%7B1%7D%7B78%7D%20%5C%5C%5C%5C%20%5Cfrac%7B31%7D%7B78%7D%20%26%20-%20%5Cfrac%7B4%7D%7B39%7D%20%26%20-%20%5Cfrac%7B11%7D%7B78%7D%20%5Cend%7Barray%7D%20%5Cright%5D%20%5Ccdot%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D5%2617%2611%5Cend%7Barray%7D%5Cright%5D)
![\left[\begin{array}{c}x_1&x_2&x_3\\\end{array}\right]=\begin{pmatrix}-\frac{2}{39}&-\frac{2}{39}&\frac{7}{39}\\ -\frac{17}{78}&\frac{11}{39}&\frac{1}{78}\\ \frac{31}{78}&-\frac{4}{39}&-\frac{11}{78}\end{pmatrix}\begin{pmatrix}5\\ 17\\ 11\end{pmatrix}=\begin{pmatrix}\frac{11}{13}\\ \frac{50}{13}\\ -\frac{17}{13}\end{pmatrix}](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Dx_1%26x_2%26x_3%5C%5C%5Cend%7Barray%7D%5Cright%5D%3D%5Cbegin%7Bpmatrix%7D-%5Cfrac%7B2%7D%7B39%7D%26-%5Cfrac%7B2%7D%7B39%7D%26%5Cfrac%7B7%7D%7B39%7D%5C%5C%20-%5Cfrac%7B17%7D%7B78%7D%26%5Cfrac%7B11%7D%7B39%7D%26%5Cfrac%7B1%7D%7B78%7D%5C%5C%20%5Cfrac%7B31%7D%7B78%7D%26-%5Cfrac%7B4%7D%7B39%7D%26-%5Cfrac%7B11%7D%7B78%7D%5Cend%7Bpmatrix%7D%5Cbegin%7Bpmatrix%7D5%5C%5C%2017%5C%5C%2011%5Cend%7Bpmatrix%7D%3D%5Cbegin%7Bpmatrix%7D%5Cfrac%7B11%7D%7B13%7D%5C%5C%20%5Cfrac%7B50%7D%7B13%7D%5C%5C%20-%5Cfrac%7B17%7D%7B13%7D%5Cend%7Bpmatrix%7D)
The solutions are
![\left[\begin{array}{c}x_1&x_2&x_3\\\end{array}\right]=\begin{pmatrix}\frac{11}{13}\\ \frac{50}{13}\\ -\frac{17}{13}\end{pmatrix}](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Dx_1%26x_2%26x_3%5C%5C%5Cend%7Barray%7D%5Cright%5D%3D%5Cbegin%7Bpmatrix%7D%5Cfrac%7B11%7D%7B13%7D%5C%5C%20%5Cfrac%7B50%7D%7B13%7D%5C%5C%20-%5Cfrac%7B17%7D%7B13%7D%5Cend%7Bpmatrix%7D)
b. To solve this system of equations
![x -y - z = 0 \\30x + 40y = 12 \\30x + 50z = 12](https://tex.z-dn.net/?f=x%20-y%20-%20z%20%3D%200%20%5C%5C30x%20%2B%2040y%20%3D%2012%20%5C%5C30x%20%2B%2050z%20%3D%2012)
The coefficient matrix is:
![A=\left[\begin{array}{ccc}1&-1&-1\\30&40&0\\30&0&50\end{array}\right]](https://tex.z-dn.net/?f=A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%26-1%26-1%5C%5C30%2640%260%5C%5C30%260%2650%5Cend%7Barray%7D%5Cright%5D)
The variable matrix is:
![X=\left[\begin{array}{c}x&y&z\\\end{array}\right]](https://tex.z-dn.net/?f=X%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Dx%26y%26z%5C%5C%5Cend%7Barray%7D%5Cright%5D)
The constant matrix is:
![B=\left[\begin{array}{c}0&12&12\\\end{array}\right]](https://tex.z-dn.net/?f=B%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D0%2612%2612%5C%5C%5Cend%7Barray%7D%5Cright%5D)
The inverse of the A matrix is
![A^{-1}=\left[ \begin{array}{ccc} \frac{20}{47} & \frac{1}{94} & \frac{2}{235} \\\\ - \frac{15}{47} & \frac{4}{235} & - \frac{3}{470} \\\\ - \frac{12}{47} & - \frac{3}{470} & \frac{7}{470} \end{array} \right]](https://tex.z-dn.net/?f=A%5E%7B-1%7D%3D%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bccc%7D%20%5Cfrac%7B20%7D%7B47%7D%20%26%20%5Cfrac%7B1%7D%7B94%7D%20%26%20%5Cfrac%7B2%7D%7B235%7D%20%5C%5C%5C%5C%20-%20%5Cfrac%7B15%7D%7B47%7D%20%26%20%5Cfrac%7B4%7D%7B235%7D%20%26%20-%20%5Cfrac%7B3%7D%7B470%7D%20%5C%5C%5C%5C%20-%20%5Cfrac%7B12%7D%7B47%7D%20%26%20-%20%5Cfrac%7B3%7D%7B470%7D%20%26%20%5Cfrac%7B7%7D%7B470%7D%20%5Cend%7Barray%7D%20%5Cright%5D)
The solutions are
![\left[\begin{array}{c}x&y&z\\\end{array}\right]=\begin{pmatrix}\frac{54}{235}\\ \frac{6}{47}\\ \frac{24}{235}\end{pmatrix}](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Dx%26y%26z%5C%5C%5Cend%7Barray%7D%5Cright%5D%3D%5Cbegin%7Bpmatrix%7D%5Cfrac%7B54%7D%7B235%7D%5C%5C%20%5Cfrac%7B6%7D%7B47%7D%5C%5C%20%5Cfrac%7B24%7D%7B235%7D%5Cend%7Bpmatrix%7D)
c. To solve this system of equations
![4x_1 + 2x_2 + x_3 + 5x_4 = 0 \\3x_1 + x_2 + 4x_3 + 7x_4 = 1\\ 2x_1 + 3x_2 + x_3 + 6x_4 = 1 \\3x_1 + x_2 + x_3 + 3x_4 = 4\\](https://tex.z-dn.net/?f=4x_1%20%2B%202x_2%20%2B%20x_3%20%2B%205x_4%20%3D%200%20%5C%5C3x_1%20%2B%20x_2%20%2B%204x_3%20%2B%207x_4%20%3D%201%5C%5C%202x_1%20%2B%203x_2%20%2B%20x_3%20%2B%206x_4%20%3D%201%20%5C%5C3x_1%20%2B%20x_2%20%2B%20x_3%20%2B%203x_4%20%3D%204%5C%5C)
The coefficient matrix is:
![A=\left[\begin{array}{cccc}4&2&1&5\\3&1&4&7\\2&3&1&6\\3&1&1&3\end{array}\right]](https://tex.z-dn.net/?f=A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D4%262%261%265%5C%5C3%261%264%267%5C%5C2%263%261%266%5C%5C3%261%261%263%5Cend%7Barray%7D%5Cright%5D)
The variable matrix is:
![X=\left[\begin{array}{c}x_1&x_2&x_3&x_4\\\end{array}\right]](https://tex.z-dn.net/?f=X%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Dx_1%26x_2%26x_3%26x_4%5C%5C%5Cend%7Barray%7D%5Cright%5D)
The constant matrix is:
![B=\left[\begin{array}{c}0&1&1&4\\\end{array}\right]](https://tex.z-dn.net/?f=B%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D0%261%261%264%5C%5C%5Cend%7Barray%7D%5Cright%5D)
The inverse of the A matrix is
![A^{-1}=\left[ \begin{array}{cccc} - \frac{1}{9} & - \frac{1}{9} & - \frac{1}{9} & \frac{2}{3} \\\\ - \frac{32}{9} & - \frac{5}{9} & \frac{13}{9} & \frac{13}{3} \\\\ - \frac{28}{9} & - \frac{1}{9} & \frac{8}{9} & \frac{11}{3} \\\\ \frac{7}{3} & \frac{1}{3} & - \frac{2}{3} & -3 \end{array} \right]](https://tex.z-dn.net/?f=A%5E%7B-1%7D%3D%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bcccc%7D%20-%20%5Cfrac%7B1%7D%7B9%7D%20%26%20-%20%5Cfrac%7B1%7D%7B9%7D%20%26%20-%20%5Cfrac%7B1%7D%7B9%7D%20%26%20%5Cfrac%7B2%7D%7B3%7D%20%5C%5C%5C%5C%20-%20%5Cfrac%7B32%7D%7B9%7D%20%26%20-%20%5Cfrac%7B5%7D%7B9%7D%20%26%20%5Cfrac%7B13%7D%7B9%7D%20%26%20%5Cfrac%7B13%7D%7B3%7D%20%5C%5C%5C%5C%20-%20%5Cfrac%7B28%7D%7B9%7D%20%26%20-%20%5Cfrac%7B1%7D%7B9%7D%20%26%20%5Cfrac%7B8%7D%7B9%7D%20%26%20%5Cfrac%7B11%7D%7B3%7D%20%5C%5C%5C%5C%20%5Cfrac%7B7%7D%7B3%7D%20%26%20%5Cfrac%7B1%7D%7B3%7D%20%26%20-%20%5Cfrac%7B2%7D%7B3%7D%20%26%20-3%20%5Cend%7Barray%7D%20%5Cright%5D)
The solutions are
![\left[\begin{array}{c}x_1&x_2&x_3&x_4\\\end{array}\right]=\begin{pmatrix}\frac{22}{9}\\ \frac{164}{9}\\ \frac{139}{9}\\ -\frac{37}{3}\end{pmatrix}](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Dx_1%26x_2%26x_3%26x_4%5C%5C%5Cend%7Barray%7D%5Cright%5D%3D%5Cbegin%7Bpmatrix%7D%5Cfrac%7B22%7D%7B9%7D%5C%5C%20%5Cfrac%7B164%7D%7B9%7D%5C%5C%20%5Cfrac%7B139%7D%7B9%7D%5C%5C%20-%5Cfrac%7B37%7D%7B3%7D%5Cend%7Bpmatrix%7D)