https://file.aiquickdraw.com/mathbot/file/173030538246819k8ja9g.jpg

To find the determinant of a 3x3 matrix, \( A \), we can use the formula:

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

where the matrix \( A \) is:

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

For the matrix given in your image:

\[
A = \begin{pmatrix} 1 & -2 & 0 \\ 3 & 1 & 5 \\ -1 & 2 & 3 \end{pmatrix}
\]

Let's identify each element in terms of \( a, b, c, d, \) etc.:

- \( a = 1 \)
- \( b = -2 \)
- \( c = 0 \)
- \( d = 3 \)
- \( e = 1 \)
- \( f = 5 \)
- \( g = -1 \)
- \( h = 2 \)
- \( i = 3 \)

### Step-by-Step Determinant Calculation:

\[
\text{det}(A) = 1 \cdot (1 \cdot 3 - 5 \cdot 2) - (-2) \cdot (3 \cdot 3 - 5 \cdot (-1)) + 0 \cdot (3 \cdot 2 - 1 \cdot (-1))
\]

Breaking it down:

1. **First term**: \( 1 \cdot (1 \cdot 3 - 5 \cdot 2) = 1 \cdot (3 - 10) = 1 \cdot (-7) = -7 \)
2. **Second term**: \( -(-2) \cdot (3 \cdot 3 - 5 \cdot (-1)) = 2 \cdot (9 + 5) = 2 \cdot 14 = 28 \)
3. **Third term**: Since the third term has \( c = 0 \), it contributes \( 0 \) to the determinant.

Adding these results:

\[
\text{det}(A) = -7 + 28 + 0 = 21
\]

### Answer:
\[
\text{det}(A) = 21
\]

Would you like further details or have any questions?

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Here are 5 related questions that expand on this topic:

1. How would the determinant calculation change if one of the entries in matrix \( A \) were different?
2. What does the determinant of a matrix tell us about the matrix?
3. How can we determine if a 3x3 matrix is invertible using its determinant?
4. How would we calculate the determinant of a 4x4 matrix?
5. Can the determinant be used to find the area or volume spanned by vectors?

**Tip:** The sign of each term in the cofactor expansion alternates. For a 3x3 matrix, it follows the pattern \( +, -, + \) for each element in the first row.

Given matrix A = [ [1, -2, 0], [3, 1, 5], [-1, 2, 3] ], find det(A).

Learn how to find the determinant of a 3x3 matrix A = [ [1, -2, 0], [3, 1, 5], [-1, 2, 3] ] using cofactor expansion. Step-by-step solution for students in Grades 10-12.

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