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Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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# Exploring Properties of Real 2x2 Matrices

This lesson focuses on the properties of real 2x2 matrices, particularly concerning matrices \( P \) and \( Q \). Here is a breakdown of the key points and conditions under consideration:

1. **P and Q are real \( 2 \times 2 \) matrices.**
   - **\( I \) = 2x2 matrix**: This refers to the identity matrix of size 2x2, which is denoted as \( I \).

**Key Conditions and Properties:**

- Select all that are true for the given matrices \( P \) and \( Q \).
- \((PQ)^2 = PQ\)
- \( P = Q^{-1} \) (Q inverse does not exist or is not defined here).
- \( P \) is invertible.
- \((P^2)^2 = P^2\)
- \((PQ^3)^3 = PQ^3 \cdot 3\)
- \( 2P = 1 \)
- \( P + 2 = 0 \)
- \( \det(P) > 0 \)
- None of these.

In exploring these properties, students should understand the basic definitions and operations for 2x2 matrices, such as multiplication, inversion, and determinants. They should also be familiar with the identity matrix and its role in matrix operations.

### Additional Notes:
- The inverse of a matrix \( Q^{-1} \), if it exists, is important in various matrix operations and can be defined only when the determinant is non-zero.
- An understanding of the determinants (\( \det(P) \)) and their significance in matrix invertibility is crucial.
- The notation \( (PQ)^2 \) refers to the matrix multiplication followed by squaring the result.

By analyzing these properties, students will gain a deeper comprehension of matrix algebra and the various characteristics that define matrix behavior in mathematical systems.
Transcribed Image Text:# Exploring Properties of Real 2x2 Matrices This lesson focuses on the properties of real 2x2 matrices, particularly concerning matrices \( P \) and \( Q \). Here is a breakdown of the key points and conditions under consideration: 1. **P and Q are real \( 2 \times 2 \) matrices.** - **\( I \) = 2x2 matrix**: This refers to the identity matrix of size 2x2, which is denoted as \( I \). **Key Conditions and Properties:** - Select all that are true for the given matrices \( P \) and \( Q \). - \((PQ)^2 = PQ\) - \( P = Q^{-1} \) (Q inverse does not exist or is not defined here). - \( P \) is invertible. - \((P^2)^2 = P^2\) - \((PQ^3)^3 = PQ^3 \cdot 3\) - \( 2P = 1 \) - \( P + 2 = 0 \) - \( \det(P) > 0 \) - None of these. In exploring these properties, students should understand the basic definitions and operations for 2x2 matrices, such as multiplication, inversion, and determinants. They should also be familiar with the identity matrix and its role in matrix operations. ### Additional Notes: - The inverse of a matrix \( Q^{-1} \), if it exists, is important in various matrix operations and can be defined only when the determinant is non-zero. - An understanding of the determinants (\( \det(P) \)) and their significance in matrix invertibility is crucial. - The notation \( (PQ)^2 \) refers to the matrix multiplication followed by squaring the result. By analyzing these properties, students will gain a deeper comprehension of matrix algebra and the various characteristics that define matrix behavior in mathematical systems.
### Express in Matrix Form

Given the system of linear equations:

\[ -x + 2y = 2 \]
\[ -x = -2 \]

To express the system in matrix form, we can write it as \( A\mathbf{x} = \mathbf{b} \), where \( A \) is the coefficient matrix, \( \mathbf{x} \) is the vector of variables, and \( \mathbf{b} \) is the constants vector.

The corresponding matrix form is:

\[ \begin{pmatrix}
-1 & 2 \\
-1 & 0 
\end{pmatrix}
\begin{pmatrix}
x \\
y 
\end{pmatrix} = 
\begin{pmatrix}
2 \\
-2 
\end{pmatrix}
\]

Here:

- \( A = \begin{pmatrix}
-1 & 2 \\
-1 & 0 
\end{pmatrix} \)  (the coefficients of \( x \) and \( y \) from each equation)
- \( \mathbf{x} = \begin{pmatrix}
x \\
y 
\end{pmatrix} \)  (the variables)
- \( \mathbf{b} = \begin{pmatrix}
2 \\
-2 
\end{pmatrix} \) (the constants from each equation)

So, the system of equations has been expressed in matrix form.
Transcribed Image Text:### Express in Matrix Form Given the system of linear equations: \[ -x + 2y = 2 \] \[ -x = -2 \] To express the system in matrix form, we can write it as \( A\mathbf{x} = \mathbf{b} \), where \( A \) is the coefficient matrix, \( \mathbf{x} \) is the vector of variables, and \( \mathbf{b} \) is the constants vector. The corresponding matrix form is: \[ \begin{pmatrix} -1 & 2 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 \\ -2 \end{pmatrix} \] Here: - \( A = \begin{pmatrix} -1 & 2 \\ -1 & 0 \end{pmatrix} \) (the coefficients of \( x \) and \( y \) from each equation) - \( \mathbf{x} = \begin{pmatrix} x \\ y \end{pmatrix} \) (the variables) - \( \mathbf{b} = \begin{pmatrix} 2 \\ -2 \end{pmatrix} \) (the constants from each equation) So, the system of equations has been expressed in matrix form.
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