9. The set of all 2 x 2 matrices of the form 07 0 b] with the standard matrix addition and scalar multiplication.
9. The set of all 2 x 2 matrices of the form 07 0 b] with the standard matrix addition and scalar multiplication.
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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Showthat the closure axioms ( 1and 6) and the
axioms for the zero vector (4) and the negative
vector (5) all hold
![**9. The set of all \(2 \times 2\) matrices of the form**
\[
\begin{bmatrix}
a & 0 \\
0 & b
\end{bmatrix}
\]
**with the standard matrix addition and scalar multiplication.**
**Explanation:**
This entry describes a specific type of \(2 \times 2\) matrix known as a diagonal matrix. In this form, all the non-diagonal elements are zero. The variables \(a\) and \(b\) can be any real numbers, representing the elements on the main diagonal.
**Operations:**
- **Matrix Addition:** When adding two matrices of this type, the resulting matrix will have diagonal elements which are the sums of the corresponding diagonal elements of the two matrices.
- **Scalar Multiplication:** When multiplying a matrix by a scalar, each element in the matrix, including both diagonals, is multiplied by that scalar. The zero elements remain unchanged.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F78fa799e-dd0d-498f-bbc7-16121e7aabca%2F4e08173e-2415-46ba-a874-22c5e80342c5%2Fn9md1fq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**9. The set of all \(2 \times 2\) matrices of the form**
\[
\begin{bmatrix}
a & 0 \\
0 & b
\end{bmatrix}
\]
**with the standard matrix addition and scalar multiplication.**
**Explanation:**
This entry describes a specific type of \(2 \times 2\) matrix known as a diagonal matrix. In this form, all the non-diagonal elements are zero. The variables \(a\) and \(b\) can be any real numbers, representing the elements on the main diagonal.
**Operations:**
- **Matrix Addition:** When adding two matrices of this type, the resulting matrix will have diagonal elements which are the sums of the corresponding diagonal elements of the two matrices.
- **Scalar Multiplication:** When multiplying a matrix by a scalar, each element in the matrix, including both diagonals, is multiplied by that scalar. The zero elements remain unchanged.
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