Question 47 on math from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.

## Q47. Given that

## and , the value of is

## (A)

## (B)

## (C)

## (D)

## Solution

To answer this question, we need to refer to **Cayley Hamilton Theorem**. This is discussed briefly in Pages 310-311 of **Introduction to Linear Algebra, Glibert Strang **(buy from Amazon.com, buy from Flipkart.com)

From the wiki entry on Cayley Hamilton theorem,

*If is a given matrix, and is the identity matrix, the characteristic polynomial of is defined as,*

*.*

*The Cayley Hamilton theorem states that substituting matrix for in this polynomial results in a zero matrix, i.e.*

**This theorem allows for to be expressed as linear combination of the lower matrix powers of .**

For a general 2×2 matrix the theorem is relatively easy to prove.

Let

The characteristic polynomial is

Substituting by matrix in the polynomial,

.

**Now, applying Cayley Hamilton theorem to the problem at hand,**

.

The characteristic polynomial is,

.

Substituting by matrix in the polynomial,

.

Alternatively, .

Finding in terms of by substituting for *,*

**Matlab example**

>> A = [-5 -3 ; 2 0]; >> A^3 ans = -65 -57 38 30 >> 19*A + 30*eye(2) ans = -65 -57 38 30

**Based on the above, the right choice is (B) ****. **

## References

[1] GATE Examination Question Papers [Previous Years] from Indian Institute of Technology, Madras http://gate.iitm.ac.in/gateqps/2012/ec.pdf

[2] **Introduction to Linear Algebra, Glibert Strang **(buy from Amazon.com, buy from Flipkart.com)

[3] wiki entry on Cayley Hamilton theorem