The Cayley-Hamilton theorem states that a square matrix satisfies its characteristic equation. That is, if is a matrix with characteristic equation then . (Here is the identity matrix of the same dimension of .)
The characteristic equation of a 2×2 matrix is a quadratic of the form , where and , where and are the eigenvalues of the matrix . Since the sum of the eigenvalues of a matrix is the sum of its diagonal elements (the trace) and the product of the eigenvalues is the determinant, we can rewrite this as
By the Cayley-Hamilton theorem,
In certain cases this formula might be an easier way to calculate the square of a 2×2 matrix than matrix multiplication, especially if the determinant is a nice number.
For example, if , we easily see that its trace is and its determinant is , so
The same formula may be applied a second time if we wish to find :
So in our example,
Finally, in case you were wondering how the Cayley-Hamilton theorem can be proved, one way is via the equation
The adjugate of an matrix is a matrix of determinants up to sign. The equation may be proved by considering the entry of each side and using the alternating sum formula for the determinant of a matrix. All that we need to be concerned with here is that if then is a polynomial of degree . We can then write the above equation as
where are constant matrices and are constant scalars.
From this we simply match coefficients of powers of of each side. (For convenience set .)
Multiplying both sides of this equation on the left by and then summing from to gives
But the left side of this equation is the characteristic equation evaluated at , and we are done.