Test by Bikram | Mock GATE | Test 2 | Question: 33

Question

Let A represent the below matrix:

$\begin{pmatrix} 1 & 0 & 3 \\ 2 & 1 & -1 \\ 1 & -1 & 1 \end{pmatrix}$

Then which among these four options are correct :

• A. $A^{3}$ $+ 3$ $A^{2}$ $+ A – 9I = 0$
• B. $A^{3}$ $- 3$ $A^{2}$ $- A + 9I = 0$
• C. $A^{3}$ $- 3$ $A^{2}$ $+ A + 9I = 0$
• D. $A^{3}$$+ 3$ $A^{2}$$- A + 9I = 0$

Answer 1

Here we Don't need to calculate the Eigen values, 

We just need the equation(cayley-Hamilton)

let K be the eigen value, then the relation AX=O we know

now

the matrix

1-k   0    3 

2    1-k   -1         =0 

1    -1   1-k 

=>(1-k){(1-k)²-1}  -0(...)  +3(2-(1-k))

=>simplifying we get

K³-3K²-K+9

now we don't need the value of K, according to cayley-hamilton, the matrix A ,matches this Equation..

so we can write

A³-3A²-A+9I=0   [I is identity matrix ]

so B is correct answer here

Answer 2

By Cayley Hamilton theorem, every square matrix satisfies its own characteristic equation so A^3 - 3A^2 - A + 9I = 0

Comments

What are the eigen values here?

---

@aman.anand

Here we Don't need to calculate the Eigen values, 

We only need the equation(cayley-Hamilton)

By Cayley Hamilton theorem, every square matrix satisfies its own characteristic equation

so A³ - 3A² - A + 9I = 0