Question

A matrix \(A\) and one of its eigenvectors are given. Find the eigenvalue of \(A\) for the given eigenvector. $$ \begin{array}{l} A=\left[\begin{array}{cc} 9 & 8 \\ -6 & -5 \end{array}\right] \\ \vec{x}=\left[\begin{array}{c} -4 \\ 3 \end{array}\right] \end{array} $$

Short answer

The eigenvalue is \( \lambda = 3 \).

Step-by-step solution

Set up the eigenvalue equation

The eigenvalue equation for a matrix \( A \) with eigenvector \( \vec{x} \) is \( A \vec{x} = \lambda \vec{x} \), where \( \lambda \) is the eigenvalue. This means we need to compute the product \( A \cdot \vec{x} \).

Compute the matrix-vector product

Calculate the product \( A \cdot \vec{x} \) by multiplying each row of matrix \( A \) with vector \( \vec{x} \). Perform the calculations:\[A \cdot \vec{x} = \begin{bmatrix} 9 & 8 \ -6 & -5 \end{bmatrix} \begin{bmatrix} -4 \ 3 \end{bmatrix}\]This gives us:\[= \begin{bmatrix} (9)(-4) + (8)(3) \ (-6)(-4) + (-5)(3) \end{bmatrix} = \begin{bmatrix} -36 + 24 \ 24 - 15 \end{bmatrix} = \begin{bmatrix} -12 \ 9 \end{bmatrix}\]

Relate the product to the eigenvalue equation

The result \( \begin{bmatrix} -12 \ 9 \end{bmatrix} \) can be written as scalar multiplication of \( \begin{bmatrix} -4 \ 3 \end{bmatrix} \). We express this relationship as:\[A \cdot \vec{x} = \lambda \begin{bmatrix} -4 \ 3 \end{bmatrix}\]From our calculation, we have:\[\begin{bmatrix} -12 \ 9 \end{bmatrix} = \lambda \begin{bmatrix} -4 \ 3 \end{bmatrix}\]

Solve for the eigenvalue \(\lambda\)

Equating each component, solve for \( \lambda \):1. From the first component: \(-12 = \lambda(-4)\). Thus, \( \lambda = \frac{-12}{-4} = 3\).2. From the second component: \(9 = \lambda(3)\). Thus, \( \lambda = \frac{9}{3} = 3\).Both components give the same eigenvalue, thus \( \lambda = 3 \) is consistent.

Key concepts

Eigenvectors

Eigenvectors are special vectors associated with a matrix that only change by a scalar factor when the matrix is applied to them through multiplication. This scalar factor is the eigenvalue. For example, if you have a matrix \( A \) and an eigenvector \( \vec{x} \), the equation \( A \vec{x} = \lambda \vec{x} \) holds, where \( \lambda \) represents the eigenvalue.
Essentially, the action of the matrix on an eigenvector results in stretching or compressing the vector by the eigenvalue, without changing its direction. Understanding the role of eigenvectors is crucial in various applications, including solving differential equations and understanding transformations in linear algebra courses.

Matrix-Vector Multiplication

Matrix-vector multiplication involves multiplying a matrix by a vector resulting in another vector. This operation combines linear combinations of the columns of the matrix, weighted by the respective components of the vector.
In our exercise example:
The matrix \( A = \begin{bmatrix} 9 & 8 \ -6 & -5 \end{bmatrix} \) is multiplied by the vector \( \vec{x} = \begin{bmatrix} -4 \ 3 \end{bmatrix} \). Each entry of the resulting vector is the sum of the products of corresponding elements of the matrix row and vector:

• The first component: \( 9 \cdot (-4) + 8 \cdot 3 = -36 + 24 = -12 \)
• The second component: \(-6 \cdot (-4) + -5 \cdot 3 = 24 - 15 = 9 \)

The result is the vector \( \begin{bmatrix} -12 \ 9 \end{bmatrix} \), showcasing the essence of matrix-vector multiplication.

Linear Algebra

Linear Algebra is the branch of mathematics that deals with vectors, vector spaces, linear transformations, and matrices. It provides the foundation for many areas of mathematics and is widely used in science and engineering.
Core concepts like matrices and eigenvectors are integral parts of linear algebra. Matrices can efficiently represent systems of linear equations, complex networks, and transformations, while eigenvectors and eigenvalues often reveal intrinsic properties of these systems.

• Linear transformations help in understanding rotations, translations, and other transformations in geometry.
• Vector spaces provide a framework to navigate multiple dimensions, which is essential for data science and machine learning.

Grasping the principles of linear algebra is pivotal for working with models and applications across various scientific domains.

Matrix Theory

Matrix theory is a central field of study within linear algebra that focuses on the properties and operations of matrices. It explores concepts such as matrix multiplication, matrix inversion, determinants, and eigenvalues/eigenvectors.
Matrix theory allows us to understand how matrix operations can describe systems of equations and transformations. By studying matrices, we gain insight into:

• The structural behavior of multiple variables and inputs.
• How transformations affect data or geometric objects through rotation, scaling, or translation.
• The stability and classifications of linear transformations that matrices can represent.

By decoding matrices' roles in mathematical modelling and real-world systems, we can effectively predict and analyze complex scenarios and phenomena.