How to find the characteristic equation of this matrix?
4 -5
2 -3
(4-x) -5
2 (-3-x)
find determinant
(4-x)(-3-x) + 10 = 0
-12 -7x + x^2 + 10 = 0
x^2 -7 x - 2 = 0
To find the characteristic equation of a matrix, you need to find the eigenvalues of the matrix. The eigenvalues can be obtained by solving the characteristic equation, which is det(A - λI) = 0, where A is the given matrix, λ is the eigenvalue, and I is the identity matrix of the same size.
Here's how you can find the characteristic equation for the given matrix:
1. Start with the given matrix:
A = | 4 -5 |
| 2 -3 |
2. Subtract λI from A:
A - λI = | 4-λ -5 |
| 2 -3-λ |
3. Calculate the determinant of A - λI:
det(A - λI) = (4-λ)(-3-λ) - (2)(-5)
= (-λ-1)(λ-2) - (-10)
= λ^2 - λ - 2 - λ -2 + 10
= λ^2 - 2λ + 8
4. Set the determinant equal to zero and solve for λ:
λ^2 - 2λ + 8 = 0
This is your characteristic equation.