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To find \( M_{23} \) and \( C_{23} \), we must compute the minor and cofactor of element \( A_{23} \) from the matrix \( A \).

Matrix \( A \) is:
\[
A = \begin{bmatrix}
6 & -1 & 1 & 7 \\
0 & 0 & -4 & 4 \\
6 & 1 & 0 & 15 \\
6 & 1 & 3 & 2
\end{bmatrix}
\]

### Minor \( M_{23} \)
To find \( M_{23} \), we delete the second row and third column from \( A \) and compute the determinant of the resulting \( 3 \times 3 \) matrix:
\[
\begin{vmatrix}
6 & -1 & 7 \\
6 & 1 & 15 \\
6 & 1 & 2
\end{vmatrix}
\]

We compute the determinant of this matrix:

\[
M_{23} = 6 \begin{vmatrix} 1 & 15 \\ 1 & 2 \end{vmatrix} - (-1) \begin{vmatrix} 6 & 15 \\ 6 & 2 \end{vmatrix} + 7 \begin{vmatrix} 6 & 1 \\ 6 & 1 \end{vmatrix}
\]

The individual \( 2 \times 2 \) determinants are:
\[
\begin{vmatrix} 1 & 15 \\ 1 & 2 \end{vmatrix} = 1(2) - 1(15) = 2 - 15 = -13
\]
\[
\begin{vmatrix} 6 & 15 \\ 6 & 2 \end{vmatrix} = 6(2) - 6(15) = 12 - 90 = -78
\]
\[
\begin{vmatrix} 6 & 1 \\ 6 & 1 \end{vmatrix} = 6(1) - 6(1) = 6 - 6 = 0
\]

Now substitute these values back:
\[
M_{23} = 6(-13) - (-1)(-78) + 7(0) = -78 - 78 + 0 = -156
\]

### Cofactor \( C_{23} \)
The cofactor is calculated using:
\[
C_{23} = (-1)^{2+3} M_{23} = (-1)^5 M_{23} = -M_{23} = 156
\]

### Final Answer:
\[
M_{23} = -156, \quad C_{23} = 156
\]

Let me know if you'd like more details or clarifications!

Here are 5 related questions:
1. How do you find the determinant of a \(3 \times 3\) matrix?
2. What is the difference between a minor and a cofactor?
3. How does the cofactor sign change based on the element's position?
4. Can you explain how to expand a determinant using minors and cofactors?
5. What role do minors and cofactors play in calculating the inverse of a matrix?

**Tip:** Always pay attention to the alternating signs when computing cofactors, especially in larger matrices!

Find M_{23} and C_{23} from matrix A = [[6, -1, 1, 7], [0, 0, -4, 4], [6, 1, 0, 15], [6, 1, 3, 2]]

This problem focuses on finding the minor M_{23} and cofactor C_{23} for element A_{23} of a 4x4 matrix. It explains how to compute the determinant of a 3x3 matrix and apply the cofactor formula, making it ideal for students studying linear algebra.

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