To calculate the determinant of the given 3x3 matrix:

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

We can use the following formula for the determinant of a 3x3 matrix \( A \):

\[
\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
\]

Where the matrix elements are:

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

Substituting the elements from the given matrix:

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

Let's calculate it step by step:

1. Calculate \( ei - fh \):

\[
ei - fh = (1)(1) - (2)(5) = 1 - 10 = -9
\]

2. Calculate \( di - fg \):

\[
di - fg = (-2)(1) - (2)(4) = -2 - 8 = -10
\]

3. Calculate \( dh - eg \):

\[
dh - eg = (-2)(5) - (1)(4) = -10 - 4 = -14
\]

Now, substitute these into the determinant formula:

\[
\text{det}(A) = 1(-9) - 3(-10) + (-1)(-14)
\]

Simplifying:

\[
\text{det}(A) = -9 + 30 + 14 = 35
\]

So, the determinant of the matrix is **35**.

Would you like more details or have any questions? Here are some possible next steps:

1. Would you like to confirm the calculation by another method?
2. Are you interested in finding the inverse of this matrix?
3. Do you need to understand how the determinant affects the properties of the matrix?
4. Would you like to solve a system of equations using this matrix?
5. Are you interested in how the determinant could apply to larger matrices?

**Tip:** The determinant can indicate whether a matrix is invertible. If the determinant is non-zero, the matrix has an inverse.

Learn how to calculate the determinant of a 3x3 matrix using the formula. This example matrix has elements 1, 3, -1, -2, 1, 2, 4, 5, 1. The determinant is calculated step-by-step, showing the application of the determinant formula and its significance in linear algebra.

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