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determine the inverse of the following matrix

The matrix \( B \) you provided is:

\[
B = \begin{pmatrix}
0 & 10 & 0 & 2 \\
0 & 5 & 0 & 1 \\
-3 & 4 & -1 & 0 \\
-8 & 7 & 1 & 6
\end{pmatrix}
\]

To find the inverse of a matrix \( B \), I will perform the following steps:

1. Verify that \( B \) is invertible by ensuring that its determinant is non-zero.
2. If the determinant is non-zero, proceed to calculate the inverse using the formula \( B^{-1} = \frac{1}{\text{det}(B)} \cdot \text{adj}(B) \), where \( \text{adj}(B) \) is the adjugate of \( B \).

Let me first calculate the determinant and proceed with finding the inverse.The determinant of matrix \( B \) is \( 0 \), which means that \( B \) is singular and does not have an inverse.

Would you like help with any other aspect of this matrix, or perhaps further details on matrix properties?

Here are 5 related questions for further exploration:
1. What are the properties of a singular matrix?
2. How can you determine if a matrix is invertible without calculating the determinant?
3. What are some methods to approximate solutions when a matrix is non-invertible?
4. How is the adjugate matrix related to the inverse?
5. What are the applications of non-invertible matrices in linear algebra?

**Tip:** A matrix with a zero determinant has linearly dependent rows or columns, which leads to it being non-invertible.

Learn how to determine whether a matrix is invertible by calculating its determinant. This problem discusses the matrix inverse formula and explains why a matrix with a zero determinant is singular. Suitable for students studying Linear Algebra.

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