307 Construct 3 different 2x2 matrices

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In summary, three different $2\times2$ matrices were constructed. The first matrix, A, has two distinct real eigenvalues of 3 and 1. The second matrix, B, has a pair of complex eigenvalues of 2-i and 2+i. The third matrix, C, has two identical real eigenvalues of 2.
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karush
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Construct 3 different $2\times2$ matrices
1. having two distinct real eigenvalues,
$A=\begin{bmatrix}
2 & -1\\
-1& 2
\end{bmatrix},\quad \left| \begin{array}{rr} 2 - \lambda & -1 \\ -1 & 2 - \lambda \end{array} \right|=\lambda^{2} - 4 \lambda + 3 \therefore \lambda_1=3\ \lambda_2=1$

2. a pair of complex eigenvalues.
$\left| \begin{array}{cc}
2 - \lambda & 1 \\
-1 & 2 - \lambda
\end{array} \right|=\lambda^{2} - 4 \lambda + 5\quad \lambda_{1}=2 - i,\lambda_{1}=2 + i$

3. two identical real eigenvalues,

ok hopefully 1 and 2 are ok

on 3 I was just going to do this $(\lambda-2)(\lambda-2)$
and go backwards to a matrix but ?
 
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For the third matrix, you can use the following:
$B=\begin{bmatrix}
2 & 0\\
0 & 2
\end{bmatrix},\quad \left| \begin{array}{rr} 2 - \lambda & 0 \\ 0 & 2 - \lambda \end{array} \right|=(2-\lambda)^2=0 \therefore \lambda_1=\lambda_2=2$.
This matrix has two identical real eigenvalues of 2.
 

FAQ: 307 Construct 3 different 2x2 matrices

1. What is a matrix?

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is used to represent and manipulate data in various fields of mathematics, science, and engineering.

2. How do you construct a 2x2 matrix?

To construct a 2x2 matrix, you need to arrange four elements in a rectangular array with two rows and two columns. The elements can be numbers, symbols, or expressions and are separated by commas. For example, the matrix [1, 2, 3, 4] is a 2x2 matrix with the elements 1, 2, 3, and 4 arranged in two rows and two columns.

3. What are the properties of a 2x2 matrix?

A 2x2 matrix has several properties, including being a square matrix (having the same number of rows and columns), being invertible (having a unique inverse), and having a determinant (a number that can be used to solve equations involving the matrix).

4. How many different 2x2 matrices can be constructed?

There are infinitely many different 2x2 matrices that can be constructed. This is because the elements of the matrix can be any numbers, symbols, or expressions, and they can be arranged in various ways.

5. What are some applications of 2x2 matrices?

2x2 matrices have many applications in mathematics, science, and engineering. They are used in linear algebra to solve systems of equations, in statistics to represent data, in physics to describe transformations and rotations, and in computer graphics to create 2D images. They also have applications in economics, finance, and other fields.

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