The image shows a mathematical problem related to matrix determinants. Here's a transcription of the text:

"Consider a matrix A: $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$ Evaluate the determinant of A. Check if the matrix is invertible."

To solve this:

1. Determinant of A:

The determinant of a 3x3 matrix $A$ with elements:

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

is calculated as:

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

For the given matrix:

$A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$

Substituting the values:

[ \text{det}(A) = 1[(5)(9) - (6)(8)] - 2[(4)(9) - (6)(7)] + 3[(4)(8) - (5)(7)] ]

Now, I'll compute the determinant.The determinant of matrix $A$ is 0.

Conclusion:

• Since the determinant is 0, the matrix $A$ is not invertible. A matrix with a zero determinant is called singular and does not have an inverse.

Would you like more details on this, or do you have any other questions?

Here are some follow-up questions to further explore this topic:

1. How do you find the inverse of a matrix if it is invertible?
2. What is the significance of a matrix having a determinant of 0?
3. How do elementary row operations affect the determinant of a matrix?
4. Can a matrix with a zero determinant still have linearly independent rows or columns?
5. What are some real-world applications of matrix determinants?

Tip: Always check the determinant of a matrix to quickly determine if it's invertible before attempting to calculate its inverse.