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Chapter 9: Problem 1

\(\mathbf{A}=\left[ \begin{array}{ll}{2} & {-4} \\ {2} & {-2}\end{array}\right]\)

Short Answer

Expert verified
The matrix \(\mathbf{A}=\left[ \begin{array}{ll}{2} & {-4} \ {2} &{-2}\end{array}\right]\) is a 2x2 matrix with elements '2', '-4', '2', '-2' arranged in two rows and two columns. The first row contains '2' and '-4' and the second row has the elements '2' and '-2'.

Step by step solution

01

Understand the Matrix Structure

The matrix \(\mathbf{A}\) is a 2x2 matrix which means it has two rows and two columns. It is represented in the format as \(\mathbf{A}=\left[ \begin{array}{ll}{a} & {b} \ {c} &{d}\end{array}\right]\), where 'a', 'b', 'c' and 'd' are elements of the matrix. In this case, the elements of the matrix \(\mathbf{A}\) are '2', '-4', '2', '-2' arranged as \(\mathbf{A}=\left[ \begin{array}{ll}{2} & {-4} \ {2} &{-2}\end{array}\right]\). We can see that the first row of the matrix has elements '2' and '-4' and the second row of the matrix has elements '2' and '-2'.
02

Identifying the Elements of the Matrix

As mentioned above, the matrix \(\mathbf{A}\) consists of four elements. These are: 'a' is 2, 'b' is -4, 'c' is 2, and 'd' is -2. These elements are placed in their appropriate positions in the matrix.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Structure
In linear algebra, comprehending the matrix structure is an essential starting point for manipulating and understanding more complex concepts. A 2x2 matrix, as in our exercise with matrix \(\mathbf{A}\), essentially resembles a grid with two rows and two columns. Visually, it looks like a small table, each slot capable of holding a number, which mathematicians call an 'element'.

The general representation of such a matrix is \(\left[ \begin{array}{cc} a & b \ c & d \end{array} \right]\), where each letter corresponds to an element occupying a specific position: 'a' and 'b' are in the first row, while 'c' and 'd' are in the second row. This layout isn't just for show; operations on matrices, such as addition, subtraction, and multiplication, rely on this orderly structure to be carried out correctly.

Understanding the matrix structure is essential as it dictates how elements interact with each other in operations and facilitates visualization when transitioning to higher dimensions in linear algebra.
Matrix Elements
The individual numbers that make up a matrix are known as matrix elements or entries. For the matrix \(\mathbf{A}\), we identify these elements distinctly as '2', '-4', '2', and '-2'. Each of these elements holds a specific position in the matrix structure:
  • The element '2' (labeled 'a' in our general format) is in the first row and first column.
  • The element '-4' ('b') is in the first row and second column.
  • The element '2' again ('c') is in the second row and first column.
  • The element '-2' ('d') is in the second row and second column.

Recognizing the position of each element is not just academic; it plays a critical role when performing matrix operations. For example, when multiplying two matrices, the element in the first row and first column of the resulting matrix is determined by combining the elements of the corresponding rows and columns of the original matrices.
Linear Algebra
Linear algebra is a branch of mathematics that deals with vectors, vector spaces, linear mappings, and systems of linear equations. It is fundamental in advancing to numerous areas of mathematics, and its concepts are widely applied in computer science, engineering, physics, and economics.

Matrices, like the 2x2 matrix \(\mathbf{A}\) in our example, are one of the key tools in linear algebra. They are useful for representing and solving linear equations, performing transformations, and describing geometrical operations. Moreover, properties of matrices, such as determinants and eigenvalues, help reveal deeper insights into linear transformations and system behaviors.

Linear algebra provides the language and framework for understanding spaces of multiple dimensions and is integral to various fields that require modeling real-world problems, from 3D graphics rendering in computer science to quantum mechanics in physics.

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Most popular questions from this chapter

\(\mathbf{A}=\left[ \begin{array}{lll}{0} & {2} & {2} \\ {2} & {0} & {2} \\\ {2} & {2} & {0}\end{array}\right]\)

\(\mathbf{A}=\left[ \begin{array}{rr}{0} & {-1} \\ {1} & {0}\end{array}\right], \quad \mathbf{f}(t)=\left[ \begin{array}{l}{t^{2}} \\\ {1}\end{array}\right]\)

(a) Show that the matrix \(\mathbf{A}=\left[ \begin{array}{lll}{2} & {1} & {6} \\ {0} & {2} & {5} \\\ {0} & {0} & {2}\end{array}\right]\) has the repeated eigenvalue \(r=2\) with multiplicity 3 and that all the eigenvectors of \(\mathbf{A}\) are of the form \(\mathbf{u}=s \operatorname{col}(1,0,0)\) (b) Use the result of part (a) to obtain a solution to the system \(\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}\) of the form \(\mathbf{x}_{1}(t)=e^{2 t} \mathbf{u}_{1} .\) (c) To obtain a second linearly independent solution to \(\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x},\) try \(\mathbf{x}_{2}(t)=t e^{2 t} \mathbf{u}_{1}+e^{2 t} \mathbf{u}_{2} .[\)Hint\(:\) Show that \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\) must satisfy \((\mathbf{A}-2 \mathbf{I}) \mathbf{u}_{1}=\mathbf{0}, \quad(\mathbf{A}-2 \mathbf{I}) \mathbf{u}_{2}=\mathbf{u}_{1} \cdot ]\) (d) To obtain a third linearly independent solution to \(\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x},\) try \(\quad \mathbf{x}_{3}(t)=\frac{t^{2}}{2} e^{2 t} \mathbf{u}_{1}+t e^{2 t} \mathbf{u}_{2}+e^{2 t} \mathbf{u}_{3}\) [Hint: Show that \(\mathbf{u}_{1}, \mathbf{u}_{2},\) and \(\mathbf{u}_{3}\) must satisfy \((\mathbf{A}-2 \mathbf{I}) \mathbf{u}_{1}=\mathbf{0}, \quad(\mathbf{A}-2 \mathbf{I}) \mathbf{u}_{2}=\mathbf{u}_{1}\) \((\mathbf{A}-2 \mathbf{I}) \mathbf{u}_{3}=\mathbf{u}_{2} . ]\) (e) Show that \((\mathbf{A}-2 \mathbf{I})^{2} \mathbf{u}_{2}=(\mathbf{A}-2 \mathbf{I})^{3} \mathbf{u}_{3}=\mathbf{0}\)

\(\mathbf{A}=\left| \begin{array}{rrrrr}{0} & {1} & {0} & {0} & {0} \\ {0} & {0} & {1} & {0} & {0} \\ {-1} & {-3} & {-3} & {0} & {0} \\ {0} & {0} & {0} & {0} & {1} \\ {0} & {0} & {0} & {-4} & {-4}\end{array}\right|\)

\(\mathbf{X}(t)=\left[ \begin{array}{ccc}{e^{t}} & {e^{-t}} & {e^{2 t}} \\\ {e^{t}} & {-e^{-t}} & {2 e^{2 t}} \\ {e^{t}} & {e^{-t}} & {4 e^{2 t}}\end{array}\right]\)
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