Cayley Hamilton Theorem: Overview, Questions, Preparation

The Cayley–Hamilton Theorem is a linear algebraic concept introduced and termed after the renowned mathematicians Arthur Cayley and William Rowan Hamilton. Cayley Hamilton Theorem explains to us that a special polynomial of a given matrix is always equal to ‘Zero’. Let us understand the theorem in detail.
The theorem states that each square matrix over a commutative ring agrees with its equation. The commutative ring includes real or complex fields.

For instance, if A is provided as n x n matrix and ln is n x n identity matrix,

Then, the distinctive polynomial of A is expressed as p (x) = det (xln – A)

Here, det refers to determinant operation; while for the ‘scalar element of the base ring’, the variable is expressed as x. The matrix entries are both linear or constant polynomials in x; the det is also n-th monic polynomial in x.

The Cayley–Hamilton theorem states that by substituting matrix, A for x in the polynomial p (x) = det (xln – A) will result in ‘Zero’ matrices, i.e., p (A) = 0
The theorem states that in the n x n matrix, when A is substituted by det (tI – A), a monic polynomial of degree n. In such case, the power of A, found by replacing the power of x (identified by recurrent matrix multiplication), the constant term of p (x), results in a power of A^0 (power stands for identity matrix)
In this theorem, A^n is allowed to be articulated as a ‘linear combination’ with A as lower matrix power. If the ring is presumed to be the field, the Cayley–Hamilton theorem would be equivalent to the statement stating that the smallest polynomial of a square matrix will be divided by its characteristic polynomial.

For example,

Consider that in an n × n matrix A over ℂ and the polynomial
p (x) = det (xln – A)

with the characteristic equation
p (A) = 0

The Cayley-Hamilton theorem states that substituting the matrix A in the characteristic polynomial results in the n × n zero matrix, i.e.
p(A) = 0n

Thus, we can say that by replacing x with matrix A, the result would be equivalent to zero. Hence here the matrix A annihilates its very own characteristic equation.

What is the use of the Cayley Hamilton Theorem?

The application of the Cayley-Hamilton Theorem is a basic commutative linear algebraic expression used for the computation of large matrices. This method is not widely used in NCERT or class 12 level problems however this concept is really important for competitive exams such as JEE and others. Students can check the Class 12 Determinants Maths Solutions for better preparation of board level exams of CBSE and other state exams.

Q: What is Cayley Hamilton Theorem?

Q: What is one of the most important applications of the Cayley Hamilton Theorem?

Q:  What is the formula of the Cayley Hamilton Theorem?

Q: Is the Cayley Hamilton Theorem applicable to all types of matrices?

Q: What does the Cayley Hamilton Theorem satisfy?