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Let's go through each part of the problem one by one.

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### Given matrices:
The matrix \( A \) is:

\[
A = \begin{pmatrix} 
1 & 0 & -1 & 2 \\
2 & 1 & 2 & -1 \\
-2 & 2 & -1 & 3 \\
3 & 4 & -1 & -2 
\end{pmatrix}
\]

### (a) Find the determinant of matrix \( B \):

The matrix \( B \) is:

\[
B = \begin{pmatrix} 
1 & 2 & -1 \\
2 & 3 & -3 \\
4 & -1 & -2 
\end{pmatrix}
\]

To find the determinant of \( B \), we will use the standard \( 3 \times 3 \) determinant formula.

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### (b) Find the inverse of matrix \( A \) using Echelon form:

The inverse of a matrix can be found by augmenting matrix \( A \) with the identity matrix and then performing row reductions to convert \( A \) to the identity matrix. The resulting augmented matrix will be the inverse of \( A \).

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### (c) Calculate the rank of matrix \( A \):

To find the rank, we will reduce matrix \( A \) to row echelon form and count the number of non-zero rows. The rank is the number of linearly independent rows.

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### (d) Write the vector \( v = (8, -1, 12, 6) \) as a linear combination of vectors \( v_1, v_2, v_3, v_4 \):

The vectors are given as:

- \( v_1 = (1, 2, -2, 3) \)
- \( v_2 = (0, 1, 2, 4) \)
- \( v_3 = (-1, 2, -1, -1) \)
- \( v_4 = (2, -1, 3, -2) \)

We need to find scalars \( c_1, c_2, c_3, c_4 \) such that:

\[
v = c_1 v_1 + c_2 v_2 + c_3 v_3 + c_4 v_4
\]

This translates to solving a system of linear equations.

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I will now proceed with the calculations for each part. Let's start with finding the determinant of matrix \( B \).The determinant of matrix \( B \) is \( -11 \).

Now, I will proceed with finding the inverse of matrix \( A \) using Echelon form.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can find the inverse of matrix \( A \) by following these steps:

1. Augment matrix \( A \) with the identity matrix.
2. Perform Gaussian elimination to reduce \( A \) to row echelon form.
3. The transformations applied to \( A \) will also transform the identity matrix into the inverse.

If you'd like, I can guide you through the manual process.

Do you want to continue solving other parts of the problem? Let me know how you'd like to proceed.

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Here are some related questions to expand your knowledge:
1. What is the relationship between the determinant and invertibility of a matrix?
2. How can the determinant help in understanding matrix transformations?
3. What is the significance of the rank of a matrix?
4. How can Gaussian elimination help in solving systems of equations?
5. Why is linear independence important when solving vector linear combinations?

**Tip**: Always check the determinant of a matrix before finding the inverse—if the determinant is zero, the matrix is singular and doesn't have an inverse.

Suppose you have given a matrix A and another matrix B. Find the determinant of matrix B, the inverse of matrix A using Echelon form, the rank of matrix A by reducing it in Echelon form, and write a vector as a linear combination of given vectors.

Learn how to find the determinant of a 3x3 matrix, invert a matrix using Echelon form, determine the rank of a matrix through reduction, and express vectors as linear combinations. This tutorial covers key concepts and formulas in advanced matrix operations suitable for university-level mathematics.

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