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To determine if the matrix \( A = \begin{pmatrix} 2 & -2 \\ 2 & 1 \end{pmatrix} \) is invertible, we can calculate its determinant. A matrix is invertible if and only if its determinant is non-zero.

For a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), the determinant is given by:

\[
\text{det}(A) = ad - bc
\]

For the given matrix \( A = \begin{pmatrix} 2 & -2 \\ 2 & 1 \end{pmatrix} \), we have:
- \( a = 2 \)
- \( b = -2 \)
- \( c = 2 \)
- \( d = 1 \)

Now, let's calculate the determinant:

\[
\text{det}(A) = (2)(1) - (-2)(2) = 2 + 4 = 6
\]

Since the determinant is 6, which is non-zero, the matrix is invertible.

Would you like more details or have any questions about this process?

Here are 5 related questions to explore further:
1. How do we calculate the inverse of a 2x2 matrix?
2. What does the determinant of a matrix tell us about its properties?
3. Can the inverse of a matrix be calculated if the determinant is zero?
4. How does the determinant change when rows or columns of a matrix are swapped?
5. What role does the determinant play in solving systems of linear equations?

**Tip**: Always check the determinant first when determining if a matrix is invertible. If it's zero, the matrix has no inverse.

Is the matrix A = (2 -2; 2 1) invertible?

Explore whether the matrix A = (2 -2; 2 1) is invertible by calculating its determinant. Learn key concepts in matrix algebra and determinant properties. This problem is suitable for students in Grades 10-12.

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