To determine if the matrix $A = \begin{pmatrix} -3 & 3 & 4 \\ -5 & -5 & 8 \\ -1 & -4 & 2 \end{pmatrix}$ is invertible, we need to check its determinant. A matrix is invertible if and only if its determinant is non-zero.

Let me calculate the determinant of this $3 \times 3$ matrix.

$\text{det}(A) = \begin{vmatrix} -3 & 3 & 4 \\ -5 & -5 & 8 \\ -1 & -4 & 2 \end{vmatrix}$

I'll compute this step-by-step.The determinant of the matrix $A$ is approximately $-1.33 \times 10^{-14}$, which is extremely close to zero. Therefore, the matrix $A$ is not invertible.

Despite the "Yes" being selected in the image, the correct answer is "No" since the determinant is essentially zero.

Would you like more details or have any questions?

Here are some related questions:

1. What happens when a matrix has a determinant of zero?
2. How can we compute the inverse of a matrix if it is invertible?
3. What is the geometric interpretation of a matrix's determinant?
4. How do matrix operations affect the determinant?
5. What role does the determinant play in systems of linear equations?

Tip: If a matrix's determinant is zero, it means the matrix maps some vectors to the zero vector, implying the transformation is not invertible.