The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation. It was independently discovered by Arthur Cayley and William Rowan Hamilton in the 1850s. Cayley proved it for 2x2 and 3x3 matrices, while Hamilton proved it for quaternion matrices. The theorem allows powers of a matrix A to be expressed as combinations of lower powers of A, which has applications in finding inverses and powers of matrices. This is useful in fields like robotics, encryption, and electrical circuits.

1. Cayley–Hamilton Theorem
What is Cayley Hamilton Theorem ?
Who found it ?
Where we use it ?

2. About Cayley-Hamilton
Arthur Cayley:(1821 – 1895) was a British Mathematician.He
helped found the modern British school of pure mathematics.As
a child, Cayley enjoyed solving complex maths problems for
amusement. He entered Trinity College, Cambridge, where he
excelled in Greek, French, German, and Italian, as well as
mathematics. He worked as a lawyer for 14 years.
He postulated the Cayley–Hamilton theorem that every square
matrix is a root of its own characteristic polynomial, and verified
it for matrices of order 2 and 3 He was the first to define the
concept of a group in the modern way as a set with a binary
operation satisfying certain laws. Formerly, when
mathematicians spoke of "groups", they had meant permutation
groups. Cayley's theorem is named in honour of Cayley.

3. About Sir William Rowan Hamilton
Sir William Rowan Hamilton: (1805 -1865) was an Irish
physicist, astronomer, and mathematician, who made
important contributions to classical mechanics, optics, and
algebra. His studies of mechanical and optical systems led
him to discover new mathematical concepts and techniques.
His best known contribution to mathematical physics is the
reformulation of Newtonian mechanics, now called
Hamiltonian mechanics. This work has proven central to the
modern study of classical field theories such as
electromagnetism, and to the development of quantum
mechanics. In pure mathematics, he is best known as the
inventor of quaternions.
Hamilton is said to have shown immense talent at a very early
age. Astronomer Bishop Dr. John Brinkley remarked of the 18-
year-old Hamilton, 'This young man, I do not say will be, but
is, the first mathematician of his age.'

4. Cayley-Hamilton Theorem
In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan
Hamilton) states that every square matrix over a commutative ring (such as the real or complex field) satisfies its own
characteristic equation.
If A is a given n×n matrix and In is the n×n identity matrix, then the characteristic polynomial of A is defined as
where det is the determinant operation and λ is a scalar element of the base ring. Since the entries of the matrix are
(linear or constant) polynomials in λ, the determinant is also an n-th order monic polynomial in λ. The Cayley–
Hamilton theorem states that substituting the matrix A for λ in this polynomial results in the zero matrix,

5. The powers of A, obtained by substitution from powers of λ, are defined by repeated matrix
multiplication; the constant term of p(λ) gives a multiple of the power A
0
, which power is defined
as the identity matrix. The theorem allows An to be expressed as a linear combination of the lower
matrix powers of A. When the ring is a field, the Cayley–Hamilton theorem is equivalent to the
statement that the minimal polynomial of a square matrix divides its characteristic polynomial.
The theorem was first proved in 1853 in terms of inverses of linear functions of quaternions, a
non-commutative ring, by Hamilton. This corresponds to the special case of certain real 4 × 4 real
or 2 × 2 complex matrices. The theorem holds for general quaternionic matrices. Cayley in 1858
stated it for 3 × 3 and smaller matrices, but only published a proof for the 2 × 2 case. The general
case was first proved by Frobenius in 1878.
Cayley-Hamilton Theorem

6. Simple Examples
1×1
matrices
2×2
matrices
As a concrete example, let
Its characteristic polynomial is given by
The Cayley–Hamilton theorem claims that, if we define
We can verify by computation that indeed,

7. A Direct Algebraic Proof
This proof uses just the kind of objects needed to formulate the Cayley–Hamilton theorem: matrices with
polynomials as entries. The matrix t In −A whose determinant is the characteristic polynomial of A is such a matrix,
and since polynomials form a commutative ring, it has an adjugate
Then, according to the right-hand fundamental relation of the adjugate, one
has
Since B is also a matrix with polynomials in t as entries, one can, for each i , collect the coefficients of ti in each
entry to form a matrix B i of numbers, such that one has
While this looks like a polynomial with matrices as coefficients, we shall not consider such a notion; it is just a way to
write a matrix with polynomial entries as a linear combination of n constant matrices, and the coefficient t i has been
written to the left of the matrix to stress this point of view.

8. Cayley - Hamilton theorem is
useful to find
 1.Inverse & power of matrix
 2.Only inverse of matrix.
 3.Only power of matrix.

9. Uses of Cayley-Hamilton
Theorem in Daily Life
Matrices are used in many fields like robotics, automation, encryption, quantum mechanics,
electrical circuits, 3D visualization in 2D etc. All places you would require powers of the matrix or
inverse.
Cayley Hamilton's theorem helps in expressing the inverse as a polynomial expression of the
matrix. Higher powers of the matrix in terms of the lower lower powers of the matrix.
The Cayley -Hamilton theorem and its generalizations have been used in control systems, electrical
circuits, systems with delays, singular systems. specially in DC circuit

10. Thank You