Question

Let \(A\) be an \(n \times n\) matrix. Given a polynomial \(p(x)=a_{0}+a_{1} x+\cdots+a_{m} x^{m}+\) we write \(p(A)=a_{0} l+a_{1} A+\cdots+a_{m} A^{m}\) For example, if \(p(x)=2-3 x+5 x^{2}\), then \(p(A)=2 l-3 A+5 A^{2}\). The characteristic polynomial of \(A\) is defined to be \(c_{A}(x)=\operatorname{det}[x I-A],\) and the Cayley. Hamilton theorem asserts that \(c_{A}(A)=0\) for any matrix \(A\). a. Verify the theorem for $$ \text { i. } A=\left[\begin{array}{rr} 3 & 2 \\ 1 & -1 \end{array}\right] \quad \text { ii. } A=\left[\begin{array}{rrr} 1 & -1 & 1 \\ 0 & 1 & 0 \\ 8 & 2 & 2 \end{array}\right] $$ b. Prove the theorem for \(A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\)

Short answer

Verify Cayley-Hamilton for the given matrices by substituting \( A \) into \( c_A(x) \) and showing it results in the zero matrix.

Step-by-step solution

Define the Characteristic Polynomial

For a given matrix \( A \), the characteristic polynomial \( c_A(x) \) is defined as \( \det(xI - A) \). For part (a)(i), let's define it for matrix \( A = \begin{bmatrix} 3 & 2 \ 1 & -1 \end{bmatrix} \).

Calculate Determinant for 2x2 Matrix

We need to determine \( \det(xI - A) \) for \( A = \begin{bmatrix} 3 & 2 \ 1 & -1 \end{bmatrix} \). The matrix \( xI - A \) becomes: \[ \begin{bmatrix} x - 3 & -2 \ -1 & x + 1 \end{bmatrix} \]. The determinant is calculated as \( (x-3)(x+1) - (-2)(-1) = x^2 - 2x - 5 \). Thus, \( c_A(x) = x^2 - 2x - 5 \).

Verify Cayley-Hamilton for 2x2 Matrix

Substitute \( A \) in the polynomial \( c_A(A) \). For \( c_A(x) = x^2 - 2x - 5 \), substitute \( A \): \( A^2 - 2A - 5I = 0 \). Compute \( A^2 = \begin{bmatrix} 3 & 2 \ 1 & -1 \end{bmatrix}^2 = \begin{bmatrix} 11 & 4 \ 1 & 0 \end{bmatrix} \); \( 2A = \begin{bmatrix} 6 & 4 \ 2 & -2 \end{bmatrix} \); \( 5I = \begin{bmatrix} 5 & 0 \ 0 & 5 \end{bmatrix} \). Calculate: \( A^2 - 2A - 5I = \begin{bmatrix} 11 & 4 \ 1 & 0 \end{bmatrix} - \begin{bmatrix} 6 & 4 \ 2 & -2 \end{bmatrix} - \begin{bmatrix} 5 & 0 \ 0 & 5 \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} \).

Evaluate Characteristic Polynomial for 3x3 Matrix

Define \( c_A(x) = \det(xI - A) \) for matrix \( A = \begin{bmatrix} 1 & -1 & 1 \ 0 & 1 & 0 \ 8 & 2 & 2 \end{bmatrix} \). Compute \( xI - A \) and find its determinant to get \( c_A(x) \).

Verify Cayley-Hamilton Theorem for 3x3 Matrix

Substitute \( A \) into the polynomial \( c_A(A) = 0 \). Using the characteristic polynomial found, compute \( A^3, A^2, \, A \), and \( I \), combine, and check if it results in the zero matrix.

General Proof for 2x2 Matrix

For matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the characteristic polynomial is \( x^2 - (a+d)x + (ad-bc) \). Substitute \( A \) in \( c_A(x) \): \( A^2 - (a+d)A + (ad-bc)I \). Compute \( A^2 = \begin{bmatrix} a & b \ c & d \end{bmatrix}^2 \) and simplify. Show it equals zero by substituting \( (a+d)A \) and \( (ad-bc)I \), finally arriving at the zero matrix.

Key concepts

Characteristic Polynomial

The characteristic polynomial is a crucial concept in matrix algebra. For any square matrix \( A \), the characteristic polynomial is defined as \( c_A(x) = \text{det}(xI - A) \). Here, \( I \) is the identity matrix, and \( \text{det} \) denotes the determinant. In essence, it is a polynomial of degree equal to the dimension of the matrix. The characteristic polynomial gives us insight into various properties of the matrix, such as eigenvalues. These eigenvalues are the roots of the polynomial.

When working with a matrix, calculating the characteristic polynomial is often the first step in solving problems related to the matrix's internal properties, like determinant and trace. Understanding the characteristic polynomial helps us dive deeper into concepts like the Cayley-Hamilton theorem, which asserts that a matrix satisfies its own characteristic polynomial. This profound result ties neatly into matrix algebra, proving useful when we analyze matrix behaviors and transformations.

Matrix Determinants

The determinant is a scalar value that can be computed from a square matrix. It is a central part of understanding the characteristic polynomial and many matrix properties. Determinants provide insight into whether a matrix has an inverse and thus plays a vital role in solving systems of linear equations.

To calculate the determinant for a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), you use the formula \( \text{det}(A) = ad - bc \). This technique helps with the computation of the characteristic polynomial \( c_A(x) \), as seen above. For 3x3 matrices or higher dimensions, the calculation becomes more complex, often requiring expansion by minors or cofactor expansion.

The determinant is a building block for the characteristic polynomial but also links to many other key areas in matrix algebra. It is critical to understand this concept fully for applications in linear transformations, volume scaling in space, and more.

Polynomial Matrices

Polynomial matrices involve matrices in which the elements are polynomials. These matrices are significant because they help generalize the regular concepts of polynomials into multi-dimensional space. For instance, considering a polynomial like \( p(x) = a_0 + a_1 x + \cdots + a_m x^m \), if we replace \( x \) with a matrix \( A \), the polynomial of the matrix is \( p(A) = a_0 I + a_1 A + \cdots + a_m A^m \).

This substitution is fundamental when employing the Cayley-Hamilton theorem, which states \( c_A(A) = 0 \). It essentially replaces each instance of \( x \) in the polynomial \( c_A(x) \) with \( A \), demonstrating that the matrix satisfies its characteristic polynomial. This application is crucial because it allows us to understand how matrices behave under polynomial operations, enabling us to solve for unknowns, reduce matrix powers, and more.

Understanding polynomial matrices helps provide a deeper grasp of matrix operations, especially when dealing with high powers of matrices in problems or real-world applications.

Matrix Algebra

Matrix algebra is a branch of mathematics focusing on matrices and operations such as addition, subtraction, multiplication, and inversion. It is foundational for understanding how systems can be represented and manipulated mathematically.

One can think of matrices as arrays of numbers that can represent data or transformations. Operations performed on matrices often solve real-world problems, such as optimizing logistics or modeling dynamic systems.

Key operations in matrix algebra include:

• *Matrix addition and subtraction*: Combine matrices of the same dimension by adding or subtracting corresponding elements.
• *Scalar multiplication*: Multiply all elements of a matrix by a constant.
• *Matrix multiplication*: A more complex operation involving the dot product of rows and columns.
• *Inversion*: Find a matrix \( A^{-1} \) such that \( AA^{-1} = I \).

The Cayley-Hamilton theorem is a result obtained from matrix algebra that states that a matrix satisfies its own characteristic polynomial, exemplifying one of the many intriguing results you can derive using this branch of mathematics. Understanding these operations ensures clarity when dealing with higher-level physics, engineering problems, and advanced mathematical concepts.