Cayley-Hamilton Theorem, a basic theorem from Linear Algebra, is not a part of JEE syllabus. But this simple result comes in handy in JEE often.
According to the Cayley-Hamilton Theorem, every square matrix A satisfies its own Characteristic Equation.
For a matrix A of size n X n, the Characteristic Equation in variable
is defined as,
,
where I is the identity matrix of size n X n.
So according to the Cayley-Hamilton Theorem, we have,
This can be illustrated with an example.
Consider,
Then,According to the Cayley-Hamilton Theorem, every square matrix A satisfies its own Characteristic Equation.
For a matrix A of size n X n, the Characteristic Equation in variable
where I is the identity matrix of size n X n.
So according to the Cayley-Hamilton Theorem, we have,
This can be illustrated with an example.
Consider,
From the Cayley-Hamilton Theorem,
Problem 1
Let
Suppose
(IIT-JEE 2005)
Solution 1
The given equation gives,
By applying the Cayley-Hamilton Theorem on A, we have,
Direct comparison of (1) and (2) gives c=-6 and d=11
thnsk for the application i ask this in my Iit jee coaching classes few days back
ReplyDeletegiven is unit vector n=(n1,n2,n3) and a real constant a. ue the cayley hamilton theorem to evaluate exp{ian. rho}
ReplyDeletehere rho is pauli matrices and n. rho=n1rho1+n2rho2+n3.rho3 and exponential of matrix is deifned by usual exponent