deduce a general formula to find THE DETERMINANT OF A 4X4 MATRIX WITH UNIQUE ELEMENTS a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q

To deduce a general formula for the determinant of a \(4 \times 4\) matrix with unique elements, we'll use cofactor expansion. This approach involves expanding the determinant along a chosen row or column and recursively calculating the determinants of the resulting smaller \(3 \times 3\) matrices.

Given a general \(4 \times 4\) matrix:

\[
A = \begin{bmatrix}
a & b & c & d \\
e & f & g & h \\
i & j & k & l \\
m & n & p & q
\end{bmatrix}
\]

### General Formula for the Determinant of a \(4 \times 4\) Matrix

The determinant of \(A\), denoted as \(\det(A)\), can be computed by expanding along the first row (or any row/column). Let's expand along the first row for simplicity:

\[
\det(A) = a \cdot \det(M_{11}) - b \cdot \det(M_{12}) + c \cdot \det(M_{13}) - d \cdot \det(M_{14}),
\]

where:

- \(M_{11}\), \(M_{12}\), \(M_{13}\), and \(M_{14}\) are the \(3 \times 3\) minors obtained by removing the respective row and column of each element in the first row.

### Determinant of Each Minor Matrix

To calculate each minor \(M_{ij}\), we need to compute a \(3 \times 3\) determinant. Let's list the minors:

1. **Minor \(M_{11}\):** Remove the first row and first column.

\[
M_{11} = \begin{bmatrix}
f & g & h \\
j & k & l \\
n & p & q
\end{bmatrix}
\]

2. **Minor \(M_{12}\):** Remove the first row and second column.

\[
M_{12} = \begin{bmatrix}
e & g & h \\
i & k & l \\
m & p & q
\end{bmatrix}
\]

3. **Minor \(M_{13}\):** Remove the first row and third column.

\[
M_{13} = \begin{bmatrix}
e & f & h \\
i & j & l \\
m & n & q
\end{bmatrix}
\]

4. **Minor \(M_{14}\):** Remove the first row and fourth column.

\[
M_{14} = \begin{bmatrix}
e & f & g \\
i & j & k \\
m & n & p
\end{bmatrix}
\]

### Determinant of a \(3 \times 3\) Matrix

The determinant of a \(3 \times 3\) matrix is given by:

\[
\det \begin{bmatrix}
x_1 & x_2 & x_3 \\
y_1 & y_2 & y_3 \\
z_1 & z_2 & z_3
\end{bmatrix} = x_1(y_2z_3 - y_3z_2) - x_2(y_1z_3 - y_3z_1) + x_3(y_1z_2 - y_2z_1).
\]

Applying this formula to each minor:

1. **\(\det(M_{11})\):**

\[
\det(M_{11}) = f(kq - lp) - g(jq - lp) + h(jp - kn)
\]

2. **\(\det(M_{12})\):**

\[
\det(M_{12}) = e(kq - lp) - g(iq - lq) + h(ip - km)
\]

3. **\(\det(M_{13})\):**

\[
\det(M_{13}) = e(jq - ln) - f(iq - lq) + h(im - jn)
\]

4. **\(\det(M_{14})\):**

\[
\det(M_{14}) = e(jp - kn) - f(ip - km) + g(im - jn)
\]

### General Formula

Substitute these minor determinants into the cofactor expansion formula:

\[
\begin{aligned}
\det(A) &= a \left( f(kq - lp) - g(jq - lp) + h(jp - kn) \right) \\
&\quad - b \left( e(kq - lp) - g(iq - lq) + h(ip - km) \right) \\
&\quad + c \left( e(jq - ln) - f(iq - lq) + h(im - jn) \right) \\
&\quad - d \left( e(jp - kn) - f(ip - km) + g(im - jn) \right).
\end{aligned}
\]

This is the general formula for the determinant of a \(4 \times 4\) matrix with unique elements \(a, b, c, d, e, f, g, h, i, j, k, l, m, n, p, q\).

### Conclusion

The formula involves a sum of products, where each term is composed of an element of the matrix and the determinant of a corresponding \(3 \times 3\) minor matrix. This general form is useful for symbolic computation or understanding the structure of matrix determinants.

Would you like to go through a specific example using this formula, or do you have any questions?

### Related Questions:
1. How do you calculate the determinant of a \(3 \times 3\) matrix using the cofactor expansion method?
2. What are the properties of determinants that can simplify their calculation?
3. How does cofactor expansion work for matrices larger than \(4 \times 4\)?
4. Can you use a different row or column for cofactor expansion to find the determinant? Does it change the result?
5. What applications do determinants have in solving systems of linear equations and in other areas of mathematics?

**Tip:** For larger matrices, using computational software or matrix reduction techniques (like row reduction) can simplify the process of finding determinants.

Explore the general formula for finding the determinant of a 4x4 matrix with unique elements. Learn the cofactor expansion method and step-by-step calculations for minors of a 4x4 matrix. Suitable for advanced high school students and anyone interested in linear algebra concepts.

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