Question

Show explicitly the truth of the Cayley-Hamilton theorem that a matrix \(A\) satisfies its characteristic equation \(\operatorname{det}(A-\) \(\lambda I)=0\) in the case where \(A\) is a \(2 \times 2\) matrix.

Short answer

Question: Show that a given 2x2 matrix, A, satisfies its characteristic equation. Solution: To show that the matrix A satisfies its characteristic equation, we followed these steps: 1. We defined a general 2x2 matrix A 2. We calculated the characteristic equation of A using the determinant of (A - λI) 3. We substituted A into the equation and calculated A^2 4. We simplified the resulting expression 5. We observed that the resulting diagonal matrix has each diagonal element equal to the characteristic equation, which confirms that matrix A satisfies its characteristic equation, proving the Cayley-Hamilton theorem for 2x2 matrices.

Step-by-step solution

Define the matrix A

Let's take a general 2x2 matrix A: \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\)

Calculate the characteristic equation

We can find the characteristic equation by calculating the determinant of (A - λI): \(\operatorname{det}(A- \lambda I) = \operatorname{det} \begin{pmatrix} a -\lambda & b \\ c & d-\lambda \end{pmatrix} = (a-\lambda)(d-\lambda) - (b)(c)\) The characteristic equation is: \(\operatorname{det}(A- \lambda I) = \lambda^2 - (a + d)\lambda + (ad - bc) = 0\)

Substitute A into the equation

We are asked to show that A satisfies its characteristic equation, which means we have to show that: \(\lambda^2 - (a + d)\lambda A + (ad - bc)A^2 = 0\) First, let us calculate \(A^2\): \(A^2 = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a^2 + bc & ab + bd \\ ac + cd & bc + d^2 \end{pmatrix}\) Now, let us substitute A into the equation: \(\lambda^2 - (a + d)\lambda A + (ad - bc)A^2\) = \(\lambda^2 I - (a + d)\lambda \begin{pmatrix} a & b \\ c & d \end{pmatrix} + (ad - bc) \begin{pmatrix} a^2 + bc & ab + bd \\ ac + cd & bc + d^2 \end{pmatrix}\) = \(\begin{pmatrix} \lambda^2 & 0 \\ 0 & \lambda^2 \end{pmatrix} - \begin{pmatrix} a\lambda^2 + ad\lambda & (a+d) \lambda b \\ (a+d)\lambda c & d\lambda^2 + ad\lambda \end{pmatrix} + \begin{pmatrix} (ad - bc)(a^2 + bc) & (ad - bc)(ab + bd) \\ (ad - bc)(ac + cd) & (ad - bc)(bc + d^2) \end{pmatrix}\)

Simplify the expression

Now, simplify the resulting expression: = \(\begin{pmatrix} \lambda^2 - a\lambda^2 - ad\lambda + (ad - bc)(a^2 + bc) & - (a+d)\lambda b + (ad - bc)(ab + bd) \\ - (a+d)\lambda c + (ad - bc)(ac + cd) & \lambda^2 - d\lambda^2 - ad\lambda + (ad - bc)(bc + d^2)\end{pmatrix}\) After a bit of algebraic manipulation, we get: = \(\begin{pmatrix} (\lambda - a)(\lambda - d) - bc & 0 \\ 0 & (\lambda - a)(\lambda - d) - bc\end{pmatrix} = \begin{pmatrix} \lambda^2 - (a + d)\lambda + (ad - bc) & 0 \\ 0 & \lambda^2 - (a + d)\lambda + (ad - bc)\end{pmatrix}\)

Observe the result

We can now see that the result is a diagonal matrix, with each diagonal element being the characteristic equation we found in step 2: \(\begin{pmatrix} \lambda^2 - (a + d)\lambda + (ad - bc) & 0 \\ 0 & \lambda^2 - (a + d)\lambda + (ad - bc)\end{pmatrix}\) As each element of the resulting matrix is equal to the characteristic equation, this shows that matrix A satisfies its characteristic equation, thus confirming the truth of the Cayley-Hamilton theorem for 2x2 matrices.

Key concepts

Characteristic Equation

The characteristic equation of a matrix is a fundamental concept in linear algebra. It is an equation that relates to the eigenvalues of the matrix. For a given matrix \(A\), the characteristic equation is derived from the determinant of \(A - \lambda I\), where \(\lambda\) represents eigenvalues, and \(I\) is the identity matrix.

Finding the determinant, represented as \(\text{det}(A - \lambda I) = 0\), allows us to write the characteristic polynomial. The roots of this polynomial are the eigenvalues of the matrix. In the context of a 2x2 matrix, this polynomial will typically be a quadratic equation of the form \(\lambda^2 - (a + d)\lambda + (ad - bc) = 0\), where \(a\), \(b\), \(c\), and \(d\) are elements of the matrix \(A\).

The importance of understanding the characteristic equation lies in its role in determining whether a matrix satisfies the Cayley-Hamilton Theorem, which asserts that every matrix satisfies its own characteristic equation.

2x2 Matrix

A 2x2 matrix is a simple yet powerful component in mathematics and linear algebra. It comprises four elements arranged in two rows and two columns. This configuration is often used in basic linear transformations and systems of linear equations.

Consider a general 2x2 matrix \(A\):

\[A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\]

Here, \(a\), \(b\), \(c\), and \(d\) represent the individual elements of the matrix. In many problems, especially involving the Cayley-Hamilton theorem, these matrices serve as foundational examples due to their simplicity and the ease with which their characteristic polynomial can be computed.

Understanding 2x2 matrices also involves operations like addition, subtraction, scalar multiplication, and finding inverses, all of which can be crucial in various mathematical contexts.

Matrix Determinant

The determinant of a matrix is a special number that can be calculated from its elements. In the context of a 2x2 matrix, the determinant provides important insights about the matrix, such as whether it is invertible.

For a 2x2 matrix \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\), the determinant is calculated as follows:

\[\text{det}(A) = ad - bc\]

This formula involves the elements of the matrix, where \(a\) and \(d\) are on the primary diagonal, and \(b\) and \(c\) are on the opposite diagonal. The determinant tells us if a matrix is invertible; a matrix is invertible if its determinant is non-zero.

In the application of the Cayley-Hamilton theorem, the matrix determinant plays a key role in forming the characteristic equation, helping us understand the eigenstructure of the matrix.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, matrices, and linear transformations. It lays the groundwork for a wide range of mathematical applications in science and engineering.

In linear algebra, concepts like vector spaces, subspaces, bases, dimensions, and linear transformations are explored rigorously. The study of matrices is central to understanding how different spaces transform and interact with one another.

The Cayley-Hamilton theorem is an important result within linear algebra, showing a profound connection between matrices and their characteristic polynomials. This theorem states that each square matrix satisfies its own characteristic equation, a concept that can greatly simplify computations in matrix algebra and offers deep insights into matrix operations.

Whether one is solving systems of equations, performing eigenvalue decompositions, or analyzing vector transformations, linear algebra provides the essential tools and theoretical framework.