To solve the problem of computing the determinant of matrix $A$, the instructions are to transform $A$ into an upper-triangular matrix $B$ using row operations and then use the relationship between $\det(A)$ and $\det(B)$.

Here's a structured approach:

1. Understanding Row Operations on Determinants:

   • Row swapping changes the sign of the determinant.
   • Multiplying a row by a scalar $c$ scales the determinant by $c$.
   • Adding a multiple of one row to another does not change the determinant.

2. Transforming $A$ to an Upper-Triangular Matrix $B$:

   • Use Gaussian elimination (row operations) to get zeros below the diagonal in matrix $A$. This would result in an upper-triangular matrix $B$ where the entries below the main diagonal are all zero.

3. Expressing $\det(A)$ in terms of $\det(B)$:

   • Once $A$ is converted to $B$, the determinant of $A$ can be written as: $\det(A) = k \cdot \det(B)$ where $k$ is a factor determined by the row operations applied.

4. Calculating $\det(B)$:

   • The determinant of an upper-triangular matrix $B$ is the product of its diagonal elements.
   • However, in the given image, $B$ seems to be a zero matrix (with all entries zero). Therefore, $\det(B) = 0$.

5. Final Computation of $\det(A)$:

   • Since $\det(B) = 0$, it follows that: $\det(A) = k \cdot 0 = 0$

Thus, the determinant of $A$ is $0$.

Would you like more details on any step or have any further questions?

Related Questions

1. What are the effects of different row operations on the determinant?
2. How do you determine the factor $k$ for row operations when calculating determinants?
3. What happens to the determinant when transforming any matrix into an upper-triangular form?
4. Why does a zero row in an upper-triangular matrix result in a zero determinant?
5. How does Gaussian elimination relate to finding determinants of matrices?

Tip:

When working with matrices and row operations, remember that any matrix with a row or column of zeros has a determinant of zero!