Question

Matrix problem i really need help with?
this is the matrix
2 0 0 0 0 0
13 -2 0 0 0 0
0 0 2 0 5 61
0 0 0 1 7 17
0 0 0 0 -1 -11
0 0 0 0 0 -2

find the determinant of this matrix A carefully explaining the determinant properties you use at each stage of your calculation

Answer 1

You can compute the determinant manually by row reducing the matrix and recording the change in determinant during each operation.

Answer 2

It's a recursive problem; you're either going to have to perform some long and tedious calculations or you can use a calculator, but the concept of a matrix's determinant is fairly simple.

Answer 3

Assuming the operator is invertible we have:

\[
\begin{eqnarray*}
A^{-1}A &=& I \\
E_1E_2...E_nA &=&I
\end{eqnarray*}
\]
So we can state the following:
\[
\begin{eqnarray*}
\det(A^{-1}) = \det(E_1E_2...E_n) = \det(E_1)\det(E_2)...\det(E_n) = \frac{1}{\det(A)}
\end{eqnarray*}
\]

Answer 4

You do not need to do any tedious operations. As proven above, elementary row operations can be used to compute the determinant.

Answer 5

Just perform the necessary operations to reduce the matrix to the identity while recording the determinant of each row operation. Take the product of them and the inverse of the product is the determinant.

Answer 6

ok thanks Alchemista and across.

Answer 7

I understand all this, is there a link to actual working out as need to be shown further confused in some areas when working it out.

Answer 8

What part of the computation are you confused about?

Answer 9

I will give it another go and get back to you.

Answer 10

Thanks Alchemista