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107 LearnersLast updated on October 23, 2025
Cayley Hamilton Theorem
The Cayley-Hamilton theorem, developed by the mathematician Arthur Cayley, is an important concept in matrix algebra. This theorem states that every square matrix satisfies its characteristic equation, which is derived from its characteristic polynomial. This theorem is useful in various mathematical applications, such as finding the inverse of a matrix and computing higher powers of a matrix. We will learn more about the Cayley-Hamilton theorem in this article.
Cayley-Hamilton Theorem
The Cayley-Hamilton theorem, developed by the mathematician Arthur Cayley, is an important concept in matrix algebra. This theorem states that every square matrix satisfies its characteristic equation, which is derived from its characteristic polynomial.
This theorem is useful in various mathematical applications, such as finding the inverse of a matrix and computing higher powers of a matrix. We will learn more about the Cayley-Hamilton theorem in this article.
What is the Cayley-Hamilton Theorem?
The Cayley-Hamilton theorem states that every square matrix satisfies its characteristic equation. This equation is derived from the matrix’s characteristic polynomial. To obtain the characteristic polynomial, subtract the scalar times the identity matrix from the given matrix and then computes the determinant of the resulting matrix.
Eigenvalues are the values of , that make the polynomial equal to zero, and they are special numbers linked to the matrix. In simple words, the Cayley-Hamilton theorem says that every square matrix satisfies an equation that is formed using that very matrix.
Cayley-Hamilton Theorem Statement
The Cayley-Hamilton theorem states that if we have a square matrix made up of real or complex numbers, there is a characteristic polynomial that comes from that matrix. When we plug the matrix into this polynomial, the result will be a zero matrix. The characteristic polynomial of an n × n matrix can be found by using the formula:
p() = det(In - A)
Here is a variable.
It is the identity matrix.
A is the given matrix,
det means the determinant of that expression.
The characteristic polynomial looks like:
p() = n + an - 1n - 1 + … + a1 + a0
Here, the highest power of is n, and its coefficient is always 1. The other terms have their constants.
According to the theorem, if we replace with the matrix A, the equation becomes:
p(A) = An + an - 1An - 1 + … + a1A + a0In, we will get 0 always.
Cayley-Hamilton Theorem Formula
The Cayley-Hamilton theorem helps simplify complex calculations and can also be used to find the inverse of a matrix efficiently. The formula for the Cayley-Hamilton theorem states that, for any n × n matrix, its characteristic polynomial looks like:
p() = n + an - 1n - 1 + … + a1 + a0
The Cayley-Hamilton theorem says:
p(A) = An + an - 1An - 1 + … + a1A + a0I
Here, A represents the given square matrix.
I is the identity matrix.
Cayley-Hamilton Theorem 2 × 2
To apply the Cayley-Hamilton theorem to a 2 × 2 matrix, follow the steps given below:
Step 1: Take a 2 × 2 matrix.
Step 2: Find its characteristic equation.
Step 3: For a 2 × 2 matrix, the characteristic equation is:
2 - S1 + S0 = 0
Here, S1 = sum of the diagonal numbers (trace of the matrix),
S2 = determinant of the matrix.
Step 4: According to the Cayley-Hamilton theorem, replace with the matrix B. The equation becomes:
B2 - S1B + S0I = 0
Where I is the identity matrix of the same order.
This means if we plug the matrix into its equation, we get a zero matrix.
Cayley-Hamilton Theorem 3 × 3
For a 3 × 3 matrix, first find the characteristic polynomial, which looks like:
p() = 3 - T22 + T1 - T0 = 0
Where, T2 = sum of the diagonal numbers,
T1 = sum of all 2 × 2 minors taken from the main diagonal elements,
T0 = determinant of the matrix.
If we put the 3 × 3 matrix into its characteristic equation, the answer is always zero. Apply the Cayley-Hamilton theorem, by replacing with the matrix C, the equation becomes,
C3 - T2C2 + T1C - T0I = 0
Where I is the identity matrix of order 3.
This means that every 3 × 3 matrix satisfies its characteristic equation.
Cayley-Hamilton Theorem Proof
There are several methods to prove the Cayley-Hamilton theorem, but the easiest method is by using substitution. Let the matrix be,
A = ca db
The theorem states that:
p(A) = A2 - (a + d)A + (ad - bc)I = 0
Step 1: Find A2
Step 2: Find (a + d)A
Step 3: Find (ad - bc)I
Step 4: Combine them
Hence, the theorem is proved.
Common Mistakes and How To Avoid Them in Cayley-Hamilton Theorem
The Cayley-Hamilton theorem is an important concept in linear algebra. While applying this theorem, students often make mistakes due to misunderstanding the steps or missing key details. Given below are some of the common mistakes and the ways to avoid them.
Mistake 1
Applying the theorem to non-square matrices
Students sometimes apply the theorem to non-square matrices by mistake, which is wrong. Cayley-Hamilton theorem applies only to a square matrix.
Before starting, check whether the given matrix has an equal number of rows and columns. Always remember that only square matrices have a characteristic polynomial.
Mistake 2
Skipping zero verification
Students sometimes stop after forming the equation and do not check if the result gives the zero matrix. After forming p(A), compute the result and confirm every element is zero.
If any element is not zero, recheck the multiplication steps.
Mistake 3
Incorrect calculation of determinant and trace
Mistakes can happen when finding the trace or determinant. A wrong determinant will make the whole verification wrong. Double-check the diagonal elements for trace.
For determinants, expand carefully or use a standard method like cross-multiplication for 2 × 2 or minors for 3 × 3.
Mistake 4
Substituting the matrix before simplifying
Some students directly substitute the matrix into the characteristic equation without expanding it properly, which leads to wrong calculations.
Always write the characteristic equation clearly in terms of and then replace it with the matrix.
Mistake 5
Forgetting to multiply by the identity matrix
When applying the theorem, the constant term in the characteristic equation should be multiplied by the identity matrix. Students mistakenly write it as A2 - (trace)A + determinant instead of A2 - (trace)A + (determinant)I.
The trace of a square matrix is the sum of its diagonals. Always check the last term; if it is a number, multiply it by I because you cannot add a scalar to a matrix quickly.
Real Life Applications of Cayley-Hamilton Theorem
The Cayley-Hamilton theorem is used in areas like engineering, physics, computer graphics, and economics to solve matrix problems quickly. It also helps to make big calculations easier, like finding inverses or powers. Here are some real-life examples of the Cayley-Hamilton theorem.
- Control Systems: In control systems, the Cayley-Hamilton theorem simplifies matrix calculations for analyzing and designing systems, such as robots, cruise control, and air conditioning, by making it easier to compute matrix powers for predicting system behavior.
- Electrical Circuits: In analyzing circuits with multiple loops, this theorem helps to solve differential equations that describe how voltage and current change over time using the matrix exponential.
- Satellite Systems: In satellite trajectory prediction, the Cayley-Hamilton theorem simplifies the complex matrices used in motion modeling, making it easier to calculate accurate orbits.
Solved Examples of Cayley-Hamilton Theorem
Problem 1
Verify the Cayley-Hamilton theorem for A = 31 42
A2 - 5A - 2I = 00 00
Explanation
Trace = 1 + 4 = 5
Determinant = (1)(4) - (2)(3) = -3
Characteristic equation: 2 - 5 - 2 = 0
Theorem: A2 - 5A - 2I = 0
A2 = 31 42 × 31 42
= 157 2210
-5A = -15-5 -20-10
-2I = 0-2 -20
Add:
157 2210 + -15-5 -20-10 + 0-2 -20 = 00 00
Problem 2
Verify the Cayley Hamilton theorem for B = 02 31
B2 - 5B + 6I = 00 00
Explanation
Trace of the given matrix is 2 + 3 = 5
Determinant of the given matrix is 2 × 3 - 0 × 1 = 6
Therefore, the characteristic equation is 2 - 5 + 6 = 0
Apply the theorem, B2 - 5B + 6I
Calculate the theorem,
Add:
Problem 3
Verify the Cayley-Hamilton theorem for
Explanation
Trace = 1 + 2 = 3
Determinant = 1 × 2 - 0 × 1 = 2
Equation = 2 - 3 + 2 = 0
Apply the theorem, C2 - 3C + 2I
Calculate the values for C2, - 3C, and 2I ,
Add:
Problem 4
Verify the Cayley-Hamilton theorem for
Explanation
Trace = 3 + 1 = 4
Determinant = 3 × 1 - 0 × 2 = 3
Equation: 2 - 4 + 3 = 0
Applying the theorem: D2 - 4D + 3I
Calculate:
Add all of them,
Problem 5
Verify the Cayley-Hamilton theorem for
Explanation
The trace is 2 + 2 = 4
The determinant is 2 × 2 - 1 × 0 = 4
Therefore, the equation becomes 2 - 4 + 4 = 0
Apply theorem: E2 - 4E + 4I
Calculate:
Add,
FAQs on Cayley-Hamilton Theorem
1.What is the Cayley-Hamilton theorem?
2.Why is the Cayley-Hamilton theorem useful?
3.Can the Cayley-Hamilton theorem be used for any matrices?
4.Who discovered the Cayley-Hamilton Theorem?
5.. Is this theorem only for real numbers?
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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