https://file.aiquickdraw.com/mathbot/file/1725815539077tq67vgfa.jpg

اختر الاجابه الصحيحه فقط

The question asks for the correct choice regarding the matrix \( A = \begin{pmatrix} 2 & 3 & 5 \\ 4 & 0 & 5 \\ 5 & 7 & -9 \end{pmatrix} \) and the value of \( \det(A) \) (the determinant of matrix \( A \)).

To solve for \( \det(A) \), let's compute the determinant using cofactor expansion along the first row:

\[
\det(A) = 2 \cdot \det\begin{pmatrix} 0 & 5 \\ 7 & -9 \end{pmatrix} - 3 \cdot \det\begin{pmatrix} 4 & 5 \\ 5 & -9 \end{pmatrix} + 5 \cdot \det\begin{pmatrix} 4 & 0 \\ 5 & 7 \end{pmatrix}
\]

Let me compute this.The determinant of matrix \( A \) is \( \det(A) = 253 \).

Thus, the correct answer for \( \det(A) \) is 253, which matches the option marked as "undefined" in the image. However, this is not correct because the determinant is clearly defined and equals 253.

Let me know if you want more details or have any questions. Here are some related questions:

1. How is the determinant of a matrix related to its invertibility?
2. Can a matrix with a zero determinant have an inverse?
3. What properties do we need for a matrix determinant to be undefined?
4. How does the size of the matrix affect the complexity of calculating the determinant?
5. How can we interpret the determinant geometrically?

**Tip:** Always double-check the signs during cofactor expansion when computing a determinant manually!

اختر الاجابة الصحيحة فقط

This problem involves calculating the determinant of a 3x3 matrix A using cofactor expansion. It explores key concepts in linear algebra such as matrix theory and determinants, providing a detailed step-by-step solution for undergraduate students.

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