GATE CS 2017 Set 2 Q52 — Eigenvalues and Eigenvectors, LU Decomposition

Let the characteristic polynomial of the $3 \times 3$ matrix $M$ be $P(\lambda) = \lambda^3 - 4\lambda^2 + a\lambda + 30$.

Since $\lambda = 2$ is an eigenvalue of $M$, it must be a root of $P(\lambda)$. Substituting $\lambda = 2$ into $P(\lambda) = 0$ gives:

Let the three eigenvalues of $M$ be $\lambda_1, \lambda_2,$ and $\lambda_3$, with $\lambda_1 = 2$. Using the properties of the roots of a polynomial:

• Sum of eigenvalues: $\lambda_1 + \lambda_2 + \lambda_3 = 4 \implies 2 + \lambda_2 + \lambda_3 = 4 \implies \lambda_2 + \lambda_3 = 2$
• Product of eigenvalues: $\lambda_1 \lambda_2 \lambda_3 = -30 \implies 2 \lambda_2 \lambda_3 = -30 \implies \lambda_2 \lambda_3 = -15$

Solving the quadratic equation $x^2 - 2x - 15 = 0$ yields the other two eigenvalues:

Thus, the eigenvalues of $M$ are $2, 5,$ and $-3$. Their absolute values are $|2| = 2$, $|5| = 5$, and $|-3| = 3$. The largest absolute value is $\mathbf{5}$.