Question

Use the Gauss-Jordan Method (1s on the diagonal and 0s everywhere else) to find the inverse of the following matrix.

Please show your work, I need it!

Answer 1

`\displaystyle \left[\begin{array}{ccc}\vphantom{\int\limits_q^B}(1)/(6)&-(1)/(6)&(1)/(3)\\\vphantom{\int\limits_q^B}-(11)/(6)&(3)/(2)&-(8)/(3)\\\vphantom{\int\limits_q^B}5&-3&5\end{array}\right]`

Explanation:

You want the inverse of the given coefficient matrix using the Gauss-Jordan elimination method.

Gauss-Jordan method

The method starts by forming an augmented matrix with the given matrix on the left, and the identity matrix on the right. Row operations are then performed that cause the matrix on the left to become the identity matrix. The matrix on the right is then the inverse of the original coefficient matrix.

For example, the first row operation is to divide the first row by the upper left coefficient, 9. This makes the first row be [1, 1/3, 1/9, 1/9, 0, 0]. This row is then used to zero the remaining 1st-column terms, by subtracting the values obtained by multiplying this by the first-column coefficients.

In the first attachment, all rows are replaced at each step, so there is not a separate step for making the diagonal term be 1. The "ReplacePart" function is what does the replacement of the designated row.

The second attachment shows a calculator computes the same result.

__

Additional comment

The number of math operations is quite large, so they are not all shown here. Rather, we have shown the result at each step of creating one column of the desired identity matrix in the first three columns.

If the starting matrix is further augmented by the constants on the right side of the matrix equation, the end result will show the solution to the equation(s). The second attachment shows the solution is ...

(a, b, c) = (-1/3, 2, 5)