Consider the following 3x3 matrix ( A ):
\[
A = \begin{pmatrix} 2 & 3 & 1 \\ 4 & 1 & -2 \\ -1 & 5 & 3 \end{pmatrix}
\].
Find determinant, then replace the first element of A with it. Find determinant for the new A, and replace the second element of A with it. Rinse and repeat until all 9 elements are replaced. What is the determinant of the final A?

We begin by computing the determinant of the matrix \( A \):

\[
A = \begin{pmatrix} 
2 & 3 & 1 \\ 
4 & 1 & -2 \\ 
-1 & 5 & 3 
\end{pmatrix}.
\]

### Step 1: Determinant of \( A \)
The determinant of a 3x3 matrix is given by:

\[
\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg),
\]
where the matrix is:

\[
\begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix}.
\]

Substituting the elements of \( A \):

\[
\text{det}(A) = 2 \left( (1)(3) - (-2)(5) \right) - 3 \left( (4)(3) - (-2)(-1) \right) + 1 \left( (4)(5) - (1)(-1) \right).
\]

Simplifying each term:

\[
\text{det}(A) = 2 \left( 3 + 10 \right) - 3 \left( 12 - 2 \right) + 1 \left( 20 + 1 \right),
\]
\[
\text{det}(A) = 2(13) - 3(10) + 1(21),
\]
\[
\text{det}(A) = 26 - 30 + 21 = 17.
\]

Thus, the determinant of \( A \) is 17.

### Step 2: Replace the first element of \( A \)
We replace the first element (which is 2) of \( A \) with the determinant (17):

\[
A_1 = \begin{pmatrix} 
17 & 3 & 1 \\ 
4 & 1 & -2 \\ 
-1 & 5 & 3 
\end{pmatrix}.
\]

### Step 3: Determinant of \( A_1 \)
We now compute the determinant of the new matrix \( A_1 \):

\[
\text{det}(A_1) = 17 \left( (1)(3) - (-2)(5) \right) - 3 \left( (4)(3) - (-2)(-1) \right) + 1 \left( (4)(5) - (1)(-1) \right).
\]

Simplifying:

\[
\text{det}(A_1) = 17 \left( 3 + 10 \right) - 3 \left( 12 - 2 \right) + 1 \left( 20 + 1 \right),
\]
\[
\text{det}(A_1) = 17(13) - 3(10) + 1(21),
\]
\[
\text{det}(A_1) = 221 - 30 + 21 = 212.
\]

Thus, the determinant of \( A_1 \) is 212.

### Step 4: Replace the second element of \( A \)
We replace the second element (which is 3) of \( A_1 \) with the determinant (212):

\[
A_2 = \begin{pmatrix} 
17 & 212 & 1 \\ 
4 & 1 & -2 \\ 
-1 & 5 & 3 
\end{pmatrix}.
\]

### Step 5: Determinant of \( A_2 \)
Now, we compute the determinant of \( A_2 \):

\[
\text{det}(A_2) = 17 \left( (1)(3) - (-2)(5) \right) - 212 \left( (4)(3) - (-2)(-1) \right) + 1 \left( (4)(5) - (1)(-1) \right).
\]

Simplifying:

\[
\text{det}(A_2) = 17(13) - 212(10) + 1(21),
\]
\[
\text{det}(A_2) = 221 - 2120 + 21 = -1878.
\]

Thus, the determinant of \( A_2 \) is -1878.

### Step 6: Replace the third element of \( A \)
We replace the third element (which is 1) of \( A_2 \) with the determinant (-1878):

\[
A_3 = \begin{pmatrix} 
17 & 212 & -1878 \\ 
4 & 1 & -2 \\ 
-1 & 5 & 3 
\end{pmatrix}.
\]

### Step 7: Determinant of \( A_3 \)
Finally, we compute the determinant of \( A_3 \):

\[
\text{det}(A_3) = 17 \left( (1)(3) - (-2)(5) \right) - 212 \left( (4)(3) - (-2)(-1) \right) + (-1878) \left( (4)(5) - (1)(-1) \right).
\]

Simplifying:

\[
\text{det}(A_3) = 17(13) - 212(10) + (-1878)(21),
\]
\[
\text{det}(A_3) = 221 - 2120 - 39438 = -41537.
\]

Thus, the determinant of the final matrix is \( \boxed{-41537} \).

Would you like any further details or explanations?

Here are five related questions to explore:
1. How is the determinant of a 3x3 matrix calculated in general?
2. What properties of matrices change or stay the same when we replace elements like this?
3. How would changing a different element of the matrix first affect the overall result?
4. What is the significance of a determinant in the context of matrix theory?
5. How could this process be generalized to larger matrices?

**Tip:** The determinant is a scalar that can give insights into the invertibility and geometric properties of a matrix.

Consider the following 3x3 matrix (A): A = \begin{pmatrix} 2 & 3 & 1 \\ 4 & 1 & -2 \\ -1 & 5 & 3 \end{pmatrix}. Find determinant, then replace the first element of A with it. Find determinant for the new A, and replace the second element of A with it. Rinse and repeat until all 9 elements are replaced. What is the determinant of the final A?

Explore how to compute the determinant of a 3x3 matrix and iteratively replace matrix elements based on determinant calculations. This step-by-step guide walks through the process of calculating determinants and transforming matrices, suitable for college-level linear algebra students.

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