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# GATE-2012 ECE Q47 (math)

by on November 13, 2012

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

## 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.combuy from Flipkart.com)

From the wiki entry on Cayley Hamilton theorem,

If $A$ is a given $\mbox{n x n}$ matrix, and $I_n$ is the $\mbox{n x n}$ identity matrix, the characteristic polynomial of $A$ is defined as,

$p(\lambda) = \det $$A-\lambda I_n$$$.

The Cayley Hamilton theorem states that substituting matrix $A$ for $\lambda$ in this polynomial results in a zero matrix, i.e.

$p(A) = 0$

This theorem allows for $A^n$ to be expressed as linear combination of the lower matrix powers of $A$.

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

Let ${A=$\begin{array}{rr}a&b\\c&d\end{array}$}$

The characteristic polynomial is

$\begin{array}{lll}p(\lambda)&=&\det$$A - \lambda I$$\\&=&\det$\begin{array}a-\lambda&b\\c&d-\lambda\end{array}$\\&=&\lambda^2 - (a+d)\lambda + $$ad-bc$$\end{array}$

Substituting by matrix $A$ in the polynomial,

$\begin{array}{lll}p(A)&=&A^2 - (a+d)A+$$ad-bc$$I\\&=&$\begin{array}(a^2+bc)&(ab+bd)\$$ac+cd)&(bc+d^2)\end{array}$-$\begin{array}(a^2+ad)&(ab+bd)\\(ac+cd)&(ad+d^2)\end{array}$+$\begin{array}(ad-bc)&0\\0&(ad-bc)\end{array}$\\&=&$\begin{array}0&0\\0&0\end{array}$\end{array}$. Now, applying Cayley Hamilton theorem to the problem at hand, ${A=$\begin{array}{rr}-5&-3\\2&0\end{array}$}$. The characteristic polynomial is, $\begin{array}{lll}p(\lambda)&=&\det\(A - \lambda I$$\\&=&\det$\begin{array}-5-\lambda&-3\\2&-\lambda\end{array}$\\&=&\lambda^2 +5\lambda+6\end{array}$.

Substituting by matrix $A$ in the polynomial,

$\begin{array}p(A)&=&A^2 +5A+6I&=&0\end{array}$.

Alternatively, $\begin{array}A^2&=&-5A-6I\end{array}$.

Finding $A^3$ in terms of $A$ by substituting for $A^2$,

$\begin{array}{lll}A^3&=&A^2A\\&=&$$-5A-6I$$A\\&=&-5A^2-6A\\&=&-5$$-5A-6I$$-6A\\&=&19A+30I\end{array}$

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) ${19A+30I}$

## 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.combuy from Flipkart.com)

[3] wiki entry on Cayley Hamilton theorem

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