https://file.aiquickdraw.com/mathbot/file/17258876947801h89ng1j.jpg

Gegeben sei die Matrix A = (■(1&2&3&1&4@0&2&1&3&3@0&0&1&2&0@1&2&3&3&4@0&4&2&4&3)) ∈ (Z/5Z)5x5
	Bestimmen Sie det A.


You are given the matrix \( A \in \mathbb{Z}_5^{5 \times 5} \), which means the matrix entries are considered modulo 5. The matrix is:

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

### Step 1: Find the determinant of \( A \) mod 5

The determinant of a matrix can be computed using cofactor expansion, or more efficiently using row reduction, considering all calculations modulo 5.

Let me compute the determinant of \( A \) modulo 5.The determinant of the matrix \( A \) modulo 5 is:

\[
\text{det}(A) = 3 \pmod{5}
\]

Would you like more details on how this was calculated or about any further steps? Here are some related questions:

1. How is matrix multiplication or determinant calculation affected when working modulo a number?
2. What are the implications if a matrix's determinant is 0 mod 5?
3. How do you row reduce a matrix in modular arithmetic?
4. Can we use the same method for larger matrices over other modular fields?
5. What is the cofactor expansion method for determinant calculation?

**Tip:** When working with matrices in modular arithmetic, remember that matrix properties (like invertibility) depend heavily on the determinant modulo the chosen number.

Gegeben sei die Matrix A = 
\(\begin{pmatrix}
1 & 2 & 3 & 1 & 4 \\
0 & 2 & 1 & 3 & 3 \\
0 & 0 & 1 & 2 & 0 \\
1 & 2 & 3 & 3 & 4 \\
0 & 4 & 2 & 4 & 3
\end{pmatrix} \in (\mathbb{Z}/5\mathbb{Z})^{5x5}\). Bestimmen Sie det A.

Learn how to calculate the determinant of a 5x5 matrix with elements in modular arithmetic (mod 5). Explore step-by-step methods using row reduction and cofactor expansion. Suitable for undergraduate students studying linear algebra.

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