Question

Starting with (A7), and using only elementary row operations, find the inverse of A = 12 1 1) 1 2 1 1 1 2) Again, you'll be graded on the correct execution of EROS

Answer 1

The inverse of the matrix A is
`A^(-1)=\left[\begin{array}{ccc}3/4&-1/4&-1/4\\-1/4&3/4&-1/4\\-1/4&-1/4&3/4\end{array}\right]`

Explanation:

We have the following matrix
`A=\left[\begin{array}{ccc}2&1&1\\1&2&1\\1&1&2\end{array}\right]`

Step 1 - Adjoin the identity matrix to the given matrix

`\left[\begin{array}ccc2&1&1&1&0&0\\1&2&1&0&1&0\\1&1&2&0&0&1\end{array}\right]`

Step 2 - Transform the matrix to the reduced row echelon form

• Row Operation 1: multiply the 1st row by 1/2

`\left[\begin{array}ccc1&1/2&1/2&1/2&0&0\\1&2&1&0&1&0\\1&1&2&0&0&1\end{array}\right]`

• Row Operation 2: add -1 times the 1st row to the 2nd row

`\left[\begin{array}ccc1&1/2&1/2&1/2&0&0\\0&3/2&1/2&-1/2&1&0\\1&1&2&0&0&1\end{array}\right]`

• Row Operation 3: add -1 times the 1st row to the 3rd row

`\left[\begin{array}ccc1&1/2&1/2&1/2&0&0\\0&3/2&1/2&-1/2&1&0\\0&1/2&3/2&-1/2&0&1\end{array}\right]`

• Row Operation 4: multiply the 2nd row by 2/3

`\left[\begin{array}ccc1&1/2&1/2&1/2&0&0\\0&1&1/3&-1/3&2/3&0\\0&1/2&3/2&-1/2&0&1\end{array}\right]`

• Row Operation 5: add -1/2 times the 2nd row to the 3rd row

`\left[\begin{array}ccc1&1/2&1/2&1/2&0&0\\0&1&1/3&-1/3&2/3&0\\0&0&4/3&-1/3&-1/3&1\end{array}\right]`

• Row Operation 6: multiply the 3rd row by 3/4

`\left[\begin{array}ccc1&1/2&1/2&1/2&0&0\\0&1&1/3&-1/3&2/3&0\\0&0&1&-1/4&-1/4&3/4\end{array}\right]`

• Row Operation 7: add -1/3 times the 3rd row to the 2nd row

`\left[\begin{array}ccc1&1/2&1/2&1/2&0&0\\0&1&0&-1/4&3/4&-1/4\\0&0&1&-1/4&-1/4&3/4\end{array}\right]`

• Row Operation 8: add -1/2 times the 3rd row to the 1st row

`\left[\begin{array}ccc1&1/2&0&5/8&1/8&-3/8\\0&1&0&-1/4&3/4&-1/4\\0&0&1&-1/4&-1/4&3/4\end{array}\right]`

• Row Operation 9: add -1/2 times the 2nd row to the 1st row

`\left[\begin{array}ccc1&0&0&3/4&-1/4&-1/4\\0&1&0&-1/4&3/4&-1/4\\0&0&1&-1/4&-1/4&3/4\end{array}\right]`

As can be seen, we have obtained the identity matrix to the left. So, we are done.

The inverse of the matrix A is
`A^(-1)=\left[\begin{array}{ccc}3/4&-1/4&-1/4\\-1/4&3/4&-1/4\\-1/4&-1/4&3/4\end{array}\right]`