GATE Civil 2022 Set 1 | Question: 15

Question

The matrix $M$ is defined as $$M = \begin{bmatrix} 1 & 3\\ 4 & 2 \end{bmatrix}$$ and has eigenvalues $5$ and $-2$. The matrix $Q$ is formed as $$Q = M^{3} - 4M^{2} - 2M$$ which of the following is/are the eigenvalues(s) of matrix $Q$?

• A. $15$
• B. $25$
• C. $-20$
• D. $-30$

Answer 1

 

If a matrix $M$ has eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$, then any matrix defined by a polynomial $Q = f(M)$ will have eigenvalues $f(\lambda_1), f(\lambda_2), \dots, f(\lambda_n)$.

In this question:

• Matrix $M$ eigenvalues: $\lambda_1 = 5$ and $\lambda_2 = -2$.

• Polynomial $Q$: $Q = M^3 - 4M^2 - 2M$.

• Eigenvalues of $Q$ ($\lambda_Q$): Substitute $\lambda_M$ into the polynomial $f(\lambda) = \lambda^3 - 4\lambda^2 - 2\lambda$.

 

Step-by-Step Calculation

1. for $\lambda_1 = 5$:

Substitute $5$ into the equation:

$$\lambda_{Q1} = (5)^3 - 4(5)^2 - 2(5)$$

$$\lambda_{Q1} = 125 - 4(25) - 10$$

$$\lambda_{Q1} = 125 - 100 - 10 = \mathbf{15}$$

2. For $\lambda_2 = -2$:

Substitute $-2$ into the equation:

$$\lambda_{Q2} = (-2)^3 - 4(-2)^2 - 2(-2)$$

$$\lambda_{Q2} = -8 - 4(4) + 4$$

$$\lambda_{Q2} = -8 - 16 + 4 = \mathbf{-20}$$

SO option $a,c$ is correct.