Question

If \(A\) is \(2 \times 2\) and has an eigenvalue \(c\) of multiplicity two, then the characteristic polynomial is \(p_{A}(\lambda)=(\lambda-c)^{2}\). Because \(p_{A}(A)=0\) (see the Cayley-Hamilton theorem), \((A-c I)^{2}=0\). Each of the \(2 \times 2\) matrices in Exercises 58 \(-61\) has an eigenvalue \(c\) of multiplicity two. Find that eigenvalue \(c\) and verify that \((A-c I)^{2}=0\). $$ \left[\begin{array}{rr} -3 & -1 \\ 1 & -1 \end{array}\right] $$

Short answer

The eigenvalue is \(c = -2\). Verified \((A-cI)^2 = 0\).

Step-by-step solution

Understanding the Matrix and Identifying the Eigenvalue

We are given the matrix:\[ A = \begin{bmatrix} -3 & -1 \ 1 & -1 \end{bmatrix} \]We need to identify the eigenvalue \(c\) which is of multiplicity two. This means the characteristic polynomial of \(A\) is \(p_A(\lambda) = (\lambda - c)^2 \).

Calculate the Characteristic Polynomial

To find the characteristic polynomial, we start with the determinant formula for eigenvalues:\[ \det(A - \lambda I) = 0 \]Here, \(I\) is the identity matrix:\[ I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]Thus, \(A - \lambda I\) becomes:\[ \begin{bmatrix} -3 - \lambda & -1 \ 1 & -1 - \lambda \end{bmatrix} \]The determinant is:\[ (-3 - \lambda)(-1 - \lambda) - (-1)(1) = \lambda^2 + 4\lambda + 3 - 1 = \lambda^2 + 4\lambda + 2 \]

Set Up the Equation for Eigenvalues

We want the characteristics polynomial to match \((\lambda - c)^2 = \lambda^2 + 4\lambda + 2\). By comparing the coefficients, we find the repeated eigenvalue:\[ c = -2 \]

Verify the Condition (A-cI)^2=0

To verify \((A - cI)^2 = 0\), first compute \(A - cI\):\[ A - (-2)I = A + 2I = \begin{bmatrix} -3 & -1 \ 1 & -1 \end{bmatrix} + \begin{bmatrix} 2 & 0 \ 0 & 2 \end{bmatrix} = \begin{bmatrix} -1 & -1 \ 1 & 1 \end{bmatrix} \]Calculate \((A-cI)^2\): \[ (A - cI) \cdot (A - cI) = \begin{bmatrix} -1 & -1 \ 1 & 1 \end{bmatrix} \cdot \begin{bmatrix} -1 & -1 \ 1 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} \]This confirms that \((A-cI)^2 = 0\).

Key concepts

Characteristic Polynomial

The characteristic polynomial is an essential concept in understanding matrix behaviors, especially in eigenvalue problems. When you have a square matrix like the one in this exercise, the characteristic polynomial is essentially a transformation of that matrix into a unique polynomial. This polynomial helps us find the eigenvalues, which are crucial for various mathematical and engineering applications.

For a matrix \(A\), the characteristic polynomial \(p_A(\lambda)\) is derived from the determinant of \(A - \lambda I\), where \(I\) is the identity matrix of the same dimension as \(A\). The formula is given by:
\[\det(A - \lambda I) = 0\]
This equation holds the eigenvalues \(\lambda\) of matrix \(A\). In our specific case, \(A\) is a \(2 \times 2\) matrix, and the characteristic polynomial simplifies to a quadratic form: \((\lambda-c)^2\), indicating a double root at the eigenvalue \(c\).

Understanding how to form and manipulate the characteristic polynomial helps you unlock the potential of matrices in solving linear algebra problems.

Cayley-Hamilton Theorem

The Cayley-Hamilton Theorem is a fascinating and powerful tool in linear algebra. It states that every square matrix satisfies its own characteristic polynomial. When this theorem is applied, it implies you can substitute the matrix itself back into its characteristic polynomial and it will equate to the zero matrix.

In our exercise, this theorem comes into play when validating that:\((A-cI)^2 = 0\).
By applying the Cayley-Hamilton Theorem, we show that the characteristic polynomial, when applied to the matrix \(A\), results in the zero matrix, hence satisfying the theorem. This is a great method to prove that our calculated eigenvalues are indeed correct and complete.

It’s important to understand that the Cayley-Hamilton Theorem is not just an abstract concept but a practical tool that confirms the consistency and reliability of eigenvalue calculations.

Matrix Algebra

Matrix algebra is a fundamental aspect of modern mathematics, with numerous applications in different scientific and engineering fields. It involves operations like addition, subtraction, scalar multiplication, and matrix multiplication. Each of these operations follows specific rules to ensure results are mathematically consistent and meaningful.

In the given exercise, we perform several key algebraic operations with a matrix \(A\) to determine its eigenvalues. One crucial operation is the matrix subtraction \(A - \lambda I\), followed by calculating the determinant to find the characteristic polynomial.

Moreover, we verify matrix identities like \((A-cI)^2 = 0\) by performing matrix multiplication. Here, understanding how to multiply matrices and recognizing that the result will be the zero matrix is essential to confirm our eigenvalue findings. This illustrates the practical aspects of matrix algebra in solving real-world problems efficiently and accurately.