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To determine whether a matrix has an inverse, we need to check its determinant. If the determinant of a matrix is non-zero, then the matrix has an inverse. If the determinant is zero, the matrix is singular and does not have an inverse.

The matrix \( A \) is given as:

\[
A = \begin{pmatrix}
2 & 0 & 4 \\
3 & -5 & -3 \\
1 & -2 & 3
\end{pmatrix}
\]

Let's calculate the determinant of \( A \) using the formula for a 3x3 matrix:

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

Where:

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

For our matrix, \( a = 2 \), \( b = 0 \), \( c = 4 \), \( d = 3 \), \( e = -5 \), \( f = -3 \), \( g = 1 \), \( h = -2 \), \( i = 3 \).

Substituting these into the determinant formula:

\[
\text{det}(A) = 2((-5)(3) - (-3)(-2)) - 0(3(3) - (-3)(1)) + 4(3(-2) - (-5)(1))
\]

\[
\text{det}(A) = 2((-15) - 6) + 4((-6) + 5)
\]

\[
\text{det}(A) = 2(-21) + 4(-1)
\]

\[
\text{det}(A) = -42 - 4 = -46
\]

Since the determinant is \(-46\), which is non-zero, the matrix **does have an inverse**.

Would you like me to proceed with finding the inverse of this matrix?

### 5 Related Questions:
1. What is the significance of the determinant in matrix operations?
2. How do you find the inverse of a 3x3 matrix step by step?
3. What properties does a matrix need to have for it to be invertible?
4. Can a matrix with a determinant of zero ever have an inverse?
5. How is the inverse of a matrix used in solving systems of equations?

### Tip:
When calculating determinants for larger matrices, breaking them down into smaller minors can simplify the process.

Find the inverse of the following matrix, if possible. Reduce all fractions to lowest terms.

Learn how to find the inverse of a 3x3 matrix with detailed steps. This problem explores key mathematical concepts like determinants and matrix inverses, suitable for advanced high school or college students.

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