How would I find the determinant of the following $4 \times 4$ matrix?

Question

\begin{pmatrix}-5&0&0&0\\ \:5&4&0&0\\ \:-1&8&3&0\\ \:-6&-3&3&1\end{pmatrix}

I know that if it was a $3 \times 3$ matrix, I could simply do the Laplace expansion on a column or row, such as the first row in the matrix above. But this does not seem entirely possible for matrices larger than a $3 \times 3$. So I am a little confused about how to solve the determinant of this one.

Any help?

Answer 1

This is a triangular matrix, the determinant is just the product of diagonal values.

Also, Laplace expansion works for arbitrary square matrices. In fact, you can prove the result that determinant is just the product of diagonals usign Laplace expansion. Try perform Laplace expansion along the first row to see this.

Answer 2

This matrix is already divided for upper triangulacja and lower triangular matrix. If you multiply all elements on main diagonal, you will have determinant. Generally speaking, If you have NxN matrix, you have to make LU decomposition. If you use Gaussian Elimination with full pivoting it will always work. If you use Gaussian Elimination with partial pivoting, it will work in most cases. After you perform Gaussian Elimination, you multiply all elements on main diagonal, and you have determinant