https://file.aiquickdraw.com/mathbot/file/1729811879344cult524p.jpg

The image shows a matrix problem labeled as "Case 2." The matrix \( B \) is given as:

\[
B = \begin{bmatrix} 
1 & 3 & 1 \\ 
2 & 5 & 3 \\ 
1 & 3 & 6 
\end{bmatrix}
\]

The tasks provided are:

1. Inverse of \( B \), denoted as \( B^{-1} \).
2. \( (2B)^{-1} \), the inverse of the matrix after being scaled by 2.
3. \( B \times B^{-1} \), the product of \( B \) and its inverse.
4. \( |B^{-1}| \), the determinant of the inverse matrix.
5. \( (B^{-1})^{-1} \), the inverse of the inverse, which should give back \( B \).
6. \( (B^T)^{-1} \), the inverse of the transpose of \( B \).

Let's begin with finding the inverse of \( B \) by calculating its determinant and adjugate matrix, then we'll solve the remaining problems.

Would you like me to compute each step and explain in detail?

Here are 5 relative questions to consider:
1. What is the formula for finding the inverse of a 3x3 matrix?
2. How does the scaling of a matrix affect its inverse?
3. What properties does the product of a matrix and its inverse have?
4. How is the determinant of an inverse matrix related to the original matrix?
5. What role does the transpose play in matrix inversion?

**Tip**: The inverse of a matrix exists only if its determinant is non-zero.

Find the inverse of matrix B and perform various operations on the inverse, as described in Case 2.

Explore the process of finding the inverse of a 3x3 matrix and performing operations like determining the inverse of a scaled matrix, the product of a matrix and its inverse, and the inverse of a transpose. This solution includes key matrix operations and concepts.

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