Question

Use the method of example 4.5 to find all of the eigenvalues λ of the matrix

a. (enter your answers as a comma-separated list.) 3 4 7 0

Answer 1

Eigenvalues are defined as those values λ (scalars) satisfying the characteristic equation:
det(A - λI) = 0

First, build the matrix A - λI =

`\left[\begin{array}{cc}3&4\\7&0\end{array}\right]` -
`\left[\begin{array}{cc}\lambda&0\\0&\lambda\end{array}\right]` =

`\left[\begin{array}{cc}3-\lambda&4\\7&-\lambda\end{array}\right]`

Then, calculate
det(A - λI) = (3 - λ)·(-λ) - (7)·(4) = -3λ + λ² - 28

Now, find the solutions to the equation:
+ λ² - 3λ - 28 = 0
λ₁₂ = [-b +/- √(b²-4ac) ] / 2a
= [3 +/- √(9 - 4·28) ] / 2
= (3 +/- 11) / 2
λ₁ = -4 and λ₂ = 7

The eigenvalues of the given matrix are λ = -4, 7