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:The inverse of the matrix A is [tex]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][/tex]Step-by-step explanation:We have the following matrix [tex]A=\left[\begin{array}{ccc}2&1&1\\1&2&1\\1&1&2\end{array}\right][/tex]Step 1 - Adjoin the identity matrix to the given matrix[tex]\left[\begin{array}{ccc|ccc}2&1&1&1&0&0\\1&2&1&0&1&0\\1&1&2&0&0&1\end{array}\right][/tex]Step 2 - Transform the matrix to the reduced row echelon formRow Operation 1: multiply the 1st row by 1/2[tex]\left[\begin{array}{ccc|ccc}1&1/2&1/2&1/2&0&0\\1&2&1&0&1&0\\1&1&2&0&0&1\end{array}\right][/tex]Row Operation 2: add -1 times the 1st row to the 2nd row[tex]\left[\begin{array}{ccc|ccc}1&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][/tex]Row Operation 3: add -1 times the 1st row to the 3rd row[tex]\left[\begin{array}{ccc|ccc}1&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][/tex]Row Operation 4: multiply the 2nd row by 2/3[tex]\left[\begin{array}{ccc|ccc}1&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][/tex]Row Operation 5: add -1/2 times the 2nd row to the 3rd row[tex]\left[\begin{array}{ccc|ccc}1&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][/tex]Row Operation 6: multiply the 3rd row by 3/4[tex]\left[\begin{array}{ccc|ccc}1&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][/tex]Row Operation 7: add -1/3 times the 3rd row to the 2nd row[tex]\left[\begin{array}{ccc|ccc}1&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][/tex]Row Operation 8: add -1/2 times the 3rd row to the 1st row[tex]\left[\begin{array}{ccc|ccc}1&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][/tex]Row Operation 9: add -1/2 times the 2nd row to the 1st row[tex]\left[\begin{array}{ccc|ccc}1&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][/tex]As can be seen, we have obtained the identity matrix to the left. So, we are done.The inverse of the matrix A is [tex]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][/tex]