Question

In Exercises \(54-57\), verify the Cayley-Hamilton theorem for the given matrix. Note: You may also want to investigate the polyvalm command. $$ \left[\begin{array}{rr} 5 & 2 \\ -6 & -2 \end{array}\right] $$

Short answer

The matrix satisfies its characteristic polynomial, verifying the Cayley-Hamilton theorem.

Step-by-step solution

Write the characteristic polynomial

For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the characteristic polynomial \( p(\lambda) \) is given by \( \lambda^2 - \text{trace}(A) \cdot \lambda + \det(A) \). Calculate the trace and determinant of \( A = \begin{bmatrix} 5 & 2 \ -6 & -2 \end{bmatrix} \). The trace is \( \text{trace}(A) = 5 + (-2) = 3 \), and the determinant is \( \det(A) = 5(-2) - 2(-6) = -10 + 12 = 2 \). Hence the characteristic polynomial is \( \lambda^2 - 3\lambda + 2 \).

Verify the polynomial with the matrix

The Cayley-Hamilton theorem states that a matrix satisfies its own characteristic polynomial. Substituting the matrix \( A \) into the polynomial \( p(A) = A^2 - 3A + 2I \), where \( I \) is the identity matrix \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \).

Calculate \( A^2 \)

To calculate \( A^2 \), multiply the matrix \( A \) by itself: \[ A^2 = \begin{bmatrix} 5 & 2 \ -6 & -2 \end{bmatrix} \cdot \begin{bmatrix} 5 & 2 \ -6 & -2 \end{bmatrix} = \begin{bmatrix} (5 \cdot 5 + 2 \cdot (-6)) & (5 \cdot 2 + 2 \cdot (-2)) \ (-6 \cdot 5 + (-2) \cdot (-6)) & (-6 \cdot 2 + (-2) \cdot (-2)) \end{bmatrix} = \begin{bmatrix} 13 & 6 \ -18 & -8 \end{bmatrix} \].

Substitute back into the polynomial

Substitute \( A^2 \) and \( 3A \) into the polynomial: \[ p(A) = A^2 - 3A + 2I = \begin{bmatrix} 13 & 6 \ -18 & -8 \end{bmatrix} - 3 \begin{bmatrix} 5 & 2 \ -6 & -2 \end{bmatrix} + 2 \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \].

Calculate \( 3A \)

Compute \( 3A \): \[ 3A = 3 \begin{bmatrix} 5 & 2 \ -6 & -2 \end{bmatrix} = \begin{bmatrix} 15 & 6 \ -18 & -6 \end{bmatrix} \].

Calculate \( 2I \)

Compute \( 2I \): \[ 2I = 2 \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 0 \ 0 & 2 \end{bmatrix} \].

Final Calculation

Perform the final calculation: \[ p(A) = \begin{bmatrix} 13 & 6 \ -18 & -8 \end{bmatrix} - \begin{bmatrix} 15 & 6 \ -18 & -6 \end{bmatrix} + \begin{bmatrix} 2 & 0 \ 0 & 2 \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} \].The result is the zero matrix, confirming that the Cayley-Hamilton theorem holds for this matrix.

Key concepts

Characteristic Polynomial

The characteristic polynomial is a special polynomial equation associated with a matrix, which plays a significant role in linear algebra, especially in the study of matrix properties. To find it, start with a square matrix, say \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \). The characteristic polynomial, \( p(\lambda) \), for a 2x2 matrix is given by:

• \( \lambda^2 - \text{trace}(A) \cdot \lambda + \det(A) \)

where \( \lambda \) represents the eigenvalue variable.

In this problem, the matrix is \( \begin{bmatrix} 5 & 2 \ -6 & -2 \end{bmatrix} \). To find the characteristic polynomial, we need to compute:

• Trace of \( A \), which is the sum of the diagonal elements, \( 5 + (-2) = 3 \).
• Determinant of \( A \), calculated as \( 5 \cdot (-2) - 2 \cdot (-6) = -10 + 12 = 2 \).

This results in the characteristic polynomial \( \lambda^2 - 3\lambda + 2 \).

Matrix Operations

Matrix operations are fundamental techniques used to manipulate matrices in order to solve problems, including verifying the Cayley-Hamilton theorem.

In our example, the steps involved in matrix operations include multiplication, scalar multiplication, and addition/subtraction.

Matrix Multiplication

To find \( A^2 \), the matrix \( A \) is multiplied by itself:

• Find each element by multiplying corresponding rows and columns.
• For the first element, calculate \((5 \cdot 5 + 2 \cdot (-6)),\) resulting in \(13\).
• Repeat for all elements to get \( A^2 = \begin{bmatrix} 13 & 6 \ -18 & -8 \end{bmatrix} \).

Scalar Multiplication

Scalar multiplication means multiplying every element in a matrix by a scalar value:

• Calculate \( 3A \) where each element of \( A \) is multiplied by 3.
• \( 3A = \begin{bmatrix} 15 & 6 \ -18 & -6 \end{bmatrix} \).

Identity Matrix

The identity matrix is a special type of matrix that acts like the number 1 in matrix multiplication. It doesn't change the other matrix when involved in a multiplication.

For a 2x2 matrix, the identity matrix \( I \) looks like this:

• \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \)

It serves a key role in verifying the Cayley-Hamilton theorem.

In this exercise, the identity matrix is used in the expression \( 2I \), where:

• Each element of \( I \) is multiplied by 2.
• This results in \( 2I = \begin{bmatrix} 2 & 0 \ 0 & 2 \end{bmatrix} \).

The identity matrix allows us to simplify the polynomial to verify it becomes the zero matrix, thus validating the theorem.

Determinant Computation

Determinants are numbers that can be calculated from square matrices and are fundamental in determining properties of matrices like invertibility and solving systems of linear equations.

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

• \( ad - bc \)

This value gives us insight into the matrix's behavior:

• If the determinant equals 0, the matrix is not invertible (or singular).
• If non-zero, the matrix is invertible.

In the example, the determinant \( \det(A) = -10 + 12 = 2 \) shows the matrix is invertible, affirming its role in the characteristic polynomial and Cayley-Hamilton verification process.