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To determine if the given matrix is invertible, we need to calculate its determinant. If the determinant is non-zero, the matrix is invertible. If the determinant is zero, the matrix is not invertible.

The matrix is:

\[
A = \begin{pmatrix} 
2 & -3 & 1 \\ 
4 & 5 & 6 \\ 
1 & 0 & 2 
\end{pmatrix}
\]

The determinant of a 3x3 matrix \( A \) can be calculated as:

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

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

For matrix \( A \), the elements are:
\[
a = 2, \, b = -3, \, c = 1, \, d = 4, \, e = 5, \, f = 6, \, g = 1, \, h = 0, \, i = 2
\]

Let me calculate the determinant.The determinant of the given matrix is \( 21 \), which is non-zero. Therefore, the matrix is invertible.

Would you like any further details or have any questions?

Here are five related questions you might find useful:
1. What does the determinant of a matrix represent geometrically?
2. How is the inverse of a 3x3 matrix calculated step by step?
3. How can we verify if a matrix is singular or not without computing the determinant?
4. What role does matrix inversion play in solving systems of linear equations?
5. How does row-reduction relate to determining whether a matrix is invertible?

**Tip:** If a matrix has a non-zero determinant, it means the matrix represents a transformation that is one-to-one and onto in linear algebra.

Is the following matrix invertible? (2, -3, 1; 4, 5, 6; 1, 0, 2)

Learn how to determine if a 3x3 matrix is invertible by calculating its determinant. In this problem, we explore the matrix (2, -3, 1; 4, 5, 6; 1, 0, 2) and determine its invertibility by finding the determinant, following the Invertibility Theorem.

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