GATE 2022 | MATHS | Q-65

Question

Let M be a $3 × 3$ real matrix such that $M^2 = 2M + 3I$. If the determinant of $M$ is $−9$, then the trace of $M$ equals._______

Answer 1

if $M^2 – 2M + 3I = 0$ then it must be a factor of the characteristic equation of the matrix M so we can say that $$\lambda^2 – 2\lambda+3 = 0 \longrightarrow \lambda_1=3,\lambda_2=-1$$ we know that $$\lambda_1\cdot\lambda_2\cdot\lambda_3=det(M) = -9 \longrightarrow\lambda_3 = \frac{9}{3\cdot-1} = -3$$

so $$trace(M) = \lambda_1+\lambda_2+\lambda_3=3+(-3)+(-1) = -1$$

notes-

1. I say factor because we know that characteristic equation of a $3 \times 3$ always has a degree of 3 whereas the given equation has degree 2
2. since degree is 3 that means there are 3 solutions so we can safely say there exists a third eigenvalue $\lambda_3$