You are given that the eigenvalues of a matrix A are 3, 2 and 2. Is A invertible? Your answer can be “yes”,“no” or “depends”. Give arguments and/or example to support your answer.


The answer is **"yes"**.

A matrix \( A \) is invertible if and only if none of its eigenvalues are zero. This is because the determinant of a matrix is the product of its eigenvalues, and a matrix is invertible if its determinant is non-zero.

In this case, the eigenvalues of \( A \) are 3, 2, and 2. Since none of these eigenvalues is zero, the determinant of \( A \) is non-zero. The determinant can be calculated as:

\[
\text{det}(A) = 3 \times 2 \times 2 = 12
\]

Since the determinant is non-zero, the matrix \( A \) is invertible.

### To summarize:
- Eigenvalues of \( A \) are 3, 2, and 2.
- The determinant of \( A \) is non-zero.
- Therefore, \( A \) is invertible.

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Do you want any more details, or have further questions?

Here are five related questions:
1. How is the determinant of a matrix related to its eigenvalues?
2. What does it mean geometrically if a matrix has an eigenvalue of 0?
3. How can you determine if a matrix is singular based on its eigenvalues?
4. How do the eigenvalues affect the condition number of a matrix?
5. How can you find the eigenvalues of a matrix from its characteristic equation?

**Tip:** The product of eigenvalues equals the determinant of a matrix, and their sum equals the trace of the matrix.

You are given that the eigenvalues of a matrix A are 3, 2 and 2. Is A invertible? Your answer can be 'yes', 'no', or 'depends'. Give arguments and/or example to support your answer.

Determine if a matrix is invertible based on its eigenvalues. This detailed solution uses the eigenvalues 3, 2, and 2 to show that the matrix is invertible, supported by the formula for the determinant and key linear algebra concepts. Ideal for undergraduate students studying linear algebra.

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