LEt A = [1 0 1, 1 0 2, -1 3 1] and B = A^-1. Find b12, the (1, 2)-entry of B.

Given that \( A = \begin{bmatrix} 1 & 0 & 1 \\ 1 & 0 & 2 \\ -1 & 3 & 1 \end{bmatrix} \), we need to find \( b_{12} \), the \((1, 2)\)-entry of the inverse matrix \( B = A^{-1} \).

### Step 1: Calculate the determinant of \( A \)
The determinant of matrix \( A \) is calculated as follows:

\[
\text{det}(A) = 1 \times \begin{vmatrix} 0 & 2 \\ 3 & 1 \end{vmatrix} - 0 \times \begin{vmatrix} 1 & 2 \\ -1 & 1 \end{vmatrix} + 1 \times \begin{vmatrix} 1 & 0 \\ -1 & 3 \end{vmatrix}
\]

First, we calculate the 2x2 determinants:

\[
\begin{vmatrix} 0 & 2 \\ 3 & 1 \end{vmatrix} = (0)(1) - (2)(3) = -6
\]

\[
\begin{vmatrix} 1 & 0 \\ -1 & 3 \end{vmatrix} = (1)(3) - (0)(-1) = 3
\]

So, the determinant is:

\[
\text{det}(A) = 1(-6) + 0 + 1(3) = -6 + 3 = -3
\]

### Step 2: Calculate the matrix of minors, cofactors, and the adjugate
Now we calculate the minors, cofactors, and then the adjugate matrix. The matrix of cofactors \( C \) is given by:

\[
C = \begin{bmatrix} c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \\ c_{31} & c_{32} & c_{33} \end{bmatrix}
\]

Where each \( c_{ij} \) is the cofactor of the element \( a_{ij} \) in matrix \( A \). The adjugate matrix is the transpose of the cofactor matrix.

For \( c_{12} \) (since we need \( b_{12} \)):

\[
c_{12} = -1 \times \begin{vmatrix} 1 & 2 \\ -1 & 1 \end{vmatrix}
\]

Calculate the determinant of the 2x2 matrix:

\[
\begin{vmatrix} 1 & 2 \\ -1 & 1 \end{vmatrix} = (1)(1) - (2)(-1) = 1 + 2 = 3
\]

So:

\[
c_{12} = -1 \times 3 = -3
\]

### Step 3: Calculate the inverse matrix \( B = A^{-1} \)
The inverse matrix \( B \) is given by:

\[
B = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)
\]

So:

\[
b_{12} = \frac{c_{12}}{\text{det}(A)} = \frac{-3}{-3} = 1
\]

### Final Answer
The \((1, 2)\)-entry of \( B \) is \( \boxed{1} \).

Would you like more details or have any further questions?

Here are some related questions you might find interesting:
1. How do you find the inverse of a 2x2 matrix?
2. What is the cofactor expansion method?
3. How is the adjugate matrix related to the inverse matrix?
4. What are some properties of invertible matrices?
5. Can every square matrix be inverted? Why or why not?
6. How does the determinant influence the invertibility of a matrix?
7. How can we verify that a matrix is its own inverse?
8. What is the significance of the identity matrix in matrix multiplication?

**Tip:** Always check the determinant first when finding an inverse, as a zero determinant means the matrix is not invertible.

Discover how to find the (1, 2)-entry b12 in an inverse matrix calculation problem using matrix algebra. This detailed solution covers determinants, cofactors, and matrix inverse formulas. Suitable for college-level students and beyond.

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