Question

The trace of the 3×3 matrix A is 16. A has an eigenvalue 7 of multiplicity 2. Find the determinant of A.

Answer 1

Answer:

Suppose your eigenvalues are x and y. your matrix A is similar to a diagonal matrix B which has it's eigenvalues on its diagonal.

Now, similar matrices have the same determinant and the same trace, thus we can get to the following equations:

2x+y=−1

x2y=45

The first one is the sum of the diagonal (we know that there are 2 unique eigenvalues thus, one of them will show up 2 times on the diagonal).

The second one is the product of the diagonal (determinant of diagonal matrix).

...y=45x2

...x=−3

if x=−3=>y=5

x2y=45 and 2x+y=−1. And that's our answer :)