Answer to Question #91408 in Calculus for Sajid

Question

Choose the correct answer.
Q.The eigen values of the matrix ■(0&a@-2&-3) are a-3 and -1. Find a.
a. a=-1
b. a=1
c. a=0
d. a=2

Answer 1

The correct answer is b) a = 1. We look for the characteristic polynomial of the given matrix

$$\begin{pmatrix} 0 & a \\ -2 & -3 \end{pmatrix}$$

To find the correct a value, we look at the characteristic polynomial roots. The characteristic polynomial can be evaluated in the following way:

$$\begin{vmatrix} 0 - \lambda & a \\ -2 & -3 - \lambda \end{vmatrix} = \lambda^2 + 3\lambda + 2a$$

Then we look for a satisfying conditions of the problem. The equation has to be correct for a-3 and -1:

$$1) (-1)^2 + 3(-1) + 2a = 0 \iff a = 1$$ $$2) (a-3)^2 + 3(a-3) +2a = 0 \iff a_{1,2} = 0, 1$$

So, the only possible answer is a = 1, a can't be equal to zero because the first equation is right if and only if a = 1.