CSIR UGC NET

Question

Let $A$ be a $3 \times 3$ real matrix. Suppose 1 and -1 are two of the three Eigen values of $A$ and 18 is one of the Eigen values of $A^2+3 A$. Then

• A. Both $A$ and $A^2+3 A$ are invertible
• B. $A^2+3 A$ is invertible but $A$ is not invertible
• C. $A$ is invertible but $A^2+3 A$ is not invertible
• D. Both $\mathrm{A}$ and $A^2+3 A$ are not invertible.

Comments

Eigen value of A:  1 , -1 and x

18 is eigen value of $A^2 +3A$==>A's 3rd Eigen value of A be either 3 or -6.

|A|=product of eigen value=1*-1*(3||6) !=0 So invertible and for |$A^2 +3A $ |=18*4*(-2) which is also non zero.

Hence Both invertible.

can  u @srestha @Shaik Masthan plz verify.

---

@Abhisek Tiwari 4

yes correct I think :)

---

Thanks.. ;)