3.1. Let 2 A = -3¹] ar 4 Find the following matrices. and B = 2 5 ==
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Matrix Operations
**Problem 3.1:**
Let
\[
A = \begin{pmatrix}
2 & -1 \\
-3 & 4
\end{pmatrix}
\]
and
\[
B = \begin{pmatrix}
2 & 1 \\
5 & -1
\end{pmatrix}.
\]
Find the following matrices.
### Explanation of Symbols
1. \( A \) and \( B \) are 2x2 matrices.
2. Each matrix element is denoted by its position in the matrix.
### Matrix A
\[
A = \begin{pmatrix}
2 & -1 \\
-3 & 4
\end{pmatrix}
\]
- The element in the first row, first column is 2.
- The element in the first row, second column is -1.
- The element in the second row, first column is -3.
- The element in the second row, second column is 4.
### Matrix B
\[
B = \begin{pmatrix}
2 & 1 \\
5 & -1
\end{pmatrix}
\]
- The element in the first row, first column is 2.
- The element in the first row, second column is 1.
- The element in the second row, first column is 5.
- The element in the second row, second column is -1.
### Task
You are required to find the following matrices based on the given \( A \) and \( B \).
---
**Note:** This task involves matrix operations such as addition, subtraction, multiplication, and finding the determinant or the inverse of the matrix. Please refer to matrix operation rules and properties for details on how to perform these calculations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d7a07fd-0eaf-4cca-88de-0657a73b7fc8%2Fdc63b112-69ea-4fe2-b803-3b4d94bc05af%2Fk6ypd46_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Matrix Operations
**Problem 3.1:**
Let
\[
A = \begin{pmatrix}
2 & -1 \\
-3 & 4
\end{pmatrix}
\]
and
\[
B = \begin{pmatrix}
2 & 1 \\
5 & -1
\end{pmatrix}.
\]
Find the following matrices.
### Explanation of Symbols
1. \( A \) and \( B \) are 2x2 matrices.
2. Each matrix element is denoted by its position in the matrix.
### Matrix A
\[
A = \begin{pmatrix}
2 & -1 \\
-3 & 4
\end{pmatrix}
\]
- The element in the first row, first column is 2.
- The element in the first row, second column is -1.
- The element in the second row, first column is -3.
- The element in the second row, second column is 4.
### Matrix B
\[
B = \begin{pmatrix}
2 & 1 \\
5 & -1
\end{pmatrix}
\]
- The element in the first row, first column is 2.
- The element in the first row, second column is 1.
- The element in the second row, first column is 5.
- The element in the second row, second column is -1.
### Task
You are required to find the following matrices based on the given \( A \) and \( B \).
---
**Note:** This task involves matrix operations such as addition, subtraction, multiplication, and finding the determinant or the inverse of the matrix. Please refer to matrix operation rules and properties for details on how to perform these calculations.
![### Matrix Transposition and Operations
In this segment, we will explore various matrix operations involving transposition, identity matrices, and polynomial expressions of matrices. Let's examine the following expressions:
#### h) \(2A^T - 3B^T\)
This expression involves the transposition of matrices \(A\) and \(B\), and scalar multiplication.
- \(A^T\) refers to the transpose of matrix \(A\).
- \(B^T\) refers to the transpose of matrix \(B\).
- The expression \(2A^T\) denotes each element of \(A^T\) multiplied by 2.
- The expression \(3B^T\) denotes each element of \(B^T\) multiplied by 3.
- Finally, these two matrices are subtracted from each other.
#### i) \((A^T)^2 - (A^2)^T\)
This expression compares the square of a transposed matrix and the transpose of a squared matrix.
- \((A^T)^2\) represents squaring the transpose of matrix \(A\).
- \((A^2)^T\) represents taking the transpose of the square of matrix \(A\).
It is important to note that matrix multiplication is generally not commutative, which affects the results of these operations.
#### j) \(2I (A - 2I)^2 I^3\), where \(I\) is the \(2 \times 2\) identity matrix
This final expression combines identity matrices and matrix polynomial expressions.
- \(I\) is specified as the \(2 \times 2\) identity matrix. The identity matrix, \(I\), has ones on the diagonal and zeros elsewhere:
\[
I = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
\]
- \(2I\) denotes each element of \(I\) multiplied by 2.
- \(A - 2I\) entails subtracting \(2I\) from the matrix \(A\).
- \((A - 2I)^2\) implies squaring the resulting matrix from the subtraction.
- \(I^3\) represents the identity matrix raised to the power of 3, which, importantly, results in \(I\). The identity matrix retains its form when](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d7a07fd-0eaf-4cca-88de-0657a73b7fc8%2Fdc63b112-69ea-4fe2-b803-3b4d94bc05af%2Fqva1evn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Matrix Transposition and Operations
In this segment, we will explore various matrix operations involving transposition, identity matrices, and polynomial expressions of matrices. Let's examine the following expressions:
#### h) \(2A^T - 3B^T\)
This expression involves the transposition of matrices \(A\) and \(B\), and scalar multiplication.
- \(A^T\) refers to the transpose of matrix \(A\).
- \(B^T\) refers to the transpose of matrix \(B\).
- The expression \(2A^T\) denotes each element of \(A^T\) multiplied by 2.
- The expression \(3B^T\) denotes each element of \(B^T\) multiplied by 3.
- Finally, these two matrices are subtracted from each other.
#### i) \((A^T)^2 - (A^2)^T\)
This expression compares the square of a transposed matrix and the transpose of a squared matrix.
- \((A^T)^2\) represents squaring the transpose of matrix \(A\).
- \((A^2)^T\) represents taking the transpose of the square of matrix \(A\).
It is important to note that matrix multiplication is generally not commutative, which affects the results of these operations.
#### j) \(2I (A - 2I)^2 I^3\), where \(I\) is the \(2 \times 2\) identity matrix
This final expression combines identity matrices and matrix polynomial expressions.
- \(I\) is specified as the \(2 \times 2\) identity matrix. The identity matrix, \(I\), has ones on the diagonal and zeros elsewhere:
\[
I = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
\]
- \(2I\) denotes each element of \(I\) multiplied by 2.
- \(A - 2I\) entails subtracting \(2I\) from the matrix \(A\).
- \((A - 2I)^2\) implies squaring the resulting matrix from the subtraction.
- \(I^3\) represents the identity matrix raised to the power of 3, which, importantly, results in \(I\). The identity matrix retains its form when
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