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To determine if a matrix is invertible, we need to calculate its determinant. If the determinant of a matrix is non-zero, then the matrix is invertible. For the given \(3 \times 3\) matrix:

\[
\begin{bmatrix} 10 & 4 & 3 \\ 2 & -12 & 6 \\ 0 & 20 & -9 \end{bmatrix}
\]

we'll calculate the determinant by expanding along the first row.

### Step-by-Step Determinant Calculation

The determinant of a \(3 \times 3\) matrix \( \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \) is calculated as:

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

For our matrix:
- \( a = 10 \), \( b = 4 \), \( c = 3 \)
- \( d = 2 \), \( e = -12 \), \( f = 6 \)
- \( g = 0 \), \( h = 20 \), \( i = -9 \)

Substitute these values into the determinant formula:

\[
\text{det} = 10((-12)(-9) - (6)(20)) - 4((2)(-9) - (6)(0)) + 3((2)(20) - (-12)(0))
\]

### Calculate Each Term

1. For the first term: \(10 \times ((-12)(-9) - (6)(20))\)
   - \((-12)(-9) = 108\)
   - \( (6)(20) = 120\)
   - So, \( 10 \times (108 - 120) = 10 \times (-12) = -120\)

2. For the second term: \(-4 \times ((2)(-9) - (6)(0))\)
   - \( (2)(-9) = -18\)
   - \( (6)(0) = 0\)
   - So, \(-4 \times (-18) = 72\)

3. For the third term: \(3 \times ((2)(20) - (-12)(0))\)
   - \( (2)(20) = 40\)
   - \((-12)(0) = 0\)
   - So, \(3 \times 40 = 120\)

### Combine the Terms

\[
\text{det} = -120 + 72 + 120 = 72
\]

Since the determinant is \(72\), which is not zero, the matrix is **invertible**.

Would you like further details on any steps, or do you have any questions?

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### Related Questions

1. What does it mean if a matrix has a determinant of zero?
2. How do you calculate the inverse of a \(3 \times 3\) matrix?
3. Can a non-square matrix be invertible?
4. How does the determinant change if we swap two rows in a matrix?
5. What is the geometric interpretation of a matrix's determinant?

### Tip

For larger matrices, using row or column expansion with the most zeros simplifies determinant calculations significantly.

Use determinants to find out if the matrix is invertible.

Learn how to determine if a 3x3 matrix is invertible by calculating its determinant. This problem covers key concepts in linear algebra, including determinants and matrix inversion, suitable for Grades 11-12.

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