Answer
$det(A)=49$
$\begin{array} \\A^{-1}=\\ \end{array}
\begin{bmatrix}-1&3&-2\\-2&0&1\\2&-1&0 \end{bmatrix}$
Work Step by Step
Step 1. List the matrix needed:
$\begin{array} \\A=\\ \end{array}
\begin{bmatrix} 1&2&3\\ 2&4&5\\2&5&6\end{bmatrix}$
Step 2. Calculate the determinant use row-1 expansion (use the formula from section 10.6):
$|A|=1(4\times6-5\times5)-2(2\times6-5\times2)+3(2\times5+4\times2)=-1-4+54=49$
Step 3. Derive $A^{-1}$ (use the procedures from section 10.5):
$\begin{array} \\AI=\\ \end{array}
\begin{bmatrix} 1&2&3&|&1&0&0\\ 2&4&5&|&0&1&0\\2&5&6&|&0&0&1 \end{bmatrix}
\begin{array} \\ \\R_3-R_2\to R_2\\R_3-2R_1\to R_3 \end{array}$
Step 4. Peform the row operations shown on the right side of the matrix:
$\begin{array} \\AI=\\ \end{array}
\begin{bmatrix} 1&2&3&|&1&0&0\\ 0&1&1&|&0&-1&1\\0&1&0&|&-2&0&1 \end{bmatrix}
\begin{array} \\ \\ \\R_2-R_3\to R_3 \end{array}$
Step 5. Peform the row operations shown on the right side of the matrix:
$\begin{array} \\AI=\\ \end{array}
\begin{bmatrix} 1&2&3&|&1&0&0\\ 0&1&1&|&0&-1&1\\0&0&1&|&2&-1&0 \end{bmatrix}
\begin{array} \\R_2-R_3\to R_2 \\ \end{array}$
Step 6. Peform the row operations shown on the right side of the matrix:
$\begin{array} \\AI=\\ \end{array}
\begin{bmatrix} 1&2&3&|&1&0&0\\ 0&1&0&|&-2&0&1\\0&0&1&|&2&-1&0 \end{bmatrix}
\begin{array} ( R_1-2R_2-3R_3\to R_1\\ \\ \\ \end{array}$
Step 7. Peform the row operations shown on the right side of the matrix:
$\begin{array} \\AI=\\ \end{array}
\begin{bmatrix} 1&0&0&|&-1&3&-2\\ 0&1&0&|&-2&0&1\\0&0&1&|&2&-1&0 \end{bmatrix}$
Step 8. Write the result as:
$\begin{array} \\A^{-1}=\\ \end{array}
\begin{bmatrix}-1&3&-2\\-2&0&1\\2&-1&0 \end{bmatrix}$
