111 400 Let A = 1 4 5 and D= 0 30 Compute AD and DA. Explain how the columns or rows of A change when A is multiplied by D on the right or on the left. Find a 3x3 matrix B, not the identity matrix or zero matrix, such that AB = BA. 15 6 0 0 2 Compute AD. AD = Compute DA. DA =D Explain how the columns or rows of A change when A is multiplied by D on the right or on the left. Choose the correct answer below. O A. Both right-multiplication (that is, multiplication on the right) and left-multiplication by the diagonal matrix D multiplies each row entry of A by the corresponding diagonal entry of D. O B. Right-multiplication (that is, multiplication on the right) by the diagonal matrix D multiplies each row of A by the corresponding diagonal entry of D. Left-multiplication by D multiplies each column of A by the corresponding diagonal entry of D. OC. Both right-multiplication (that is, multiplication on the right) and left-multiplication by the diagonal matrix D multiplies each column entry of A by the corresponding diagonal entry of D. O D. Right-multiplication (that is, multiplication on the right) by the diagonal matrix D multiplies each column of A by the corresponding diagonal entry of D. Left-multiplication by D multiplies each row of A by the corresponding diagonal entry of D. Find a 3x3 matrix B, not the identity matrix or zero matrix, such that AB = BA. Choose the correct answer below. There is only one unique solution, B = OA. (Simplify your answers.)
111 400 Let A = 1 4 5 and D= 0 30 Compute AD and DA. Explain how the columns or rows of A change when A is multiplied by D on the right or on the left. Find a 3x3 matrix B, not the identity matrix or zero matrix, such that AB = BA. 15 6 0 0 2 Compute AD. AD = Compute DA. DA =D Explain how the columns or rows of A change when A is multiplied by D on the right or on the left. Choose the correct answer below. O A. Both right-multiplication (that is, multiplication on the right) and left-multiplication by the diagonal matrix D multiplies each row entry of A by the corresponding diagonal entry of D. O B. Right-multiplication (that is, multiplication on the right) by the diagonal matrix D multiplies each row of A by the corresponding diagonal entry of D. Left-multiplication by D multiplies each column of A by the corresponding diagonal entry of D. OC. Both right-multiplication (that is, multiplication on the right) and left-multiplication by the diagonal matrix D multiplies each column entry of A by the corresponding diagonal entry of D. O D. Right-multiplication (that is, multiplication on the right) by the diagonal matrix D multiplies each column of A by the corresponding diagonal entry of D. Left-multiplication by D multiplies each row of A by the corresponding diagonal entry of D. Find a 3x3 matrix B, not the identity matrix or zero matrix, such that AB = BA. Choose the correct answer below. There is only one unique solution, B = OA. (Simplify your answers.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Matrix Multiplication and Properties of Diagonal Matrices**
Given matrices A and D:
\[ A = \begin{bmatrix} 1 & 1 & 1 \\ 4 & 5 & 6 \\ 1 & 5 & 6 \end{bmatrix},\quad D = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 2 \end{bmatrix} \]
We need to compute \( AD \) and \( DA \) and analyze how the columns or rows of \( A \) change when it is multiplied by \( D \) on the right or on the left. Additionally, we are to find a 3x3 matrix \( B \), not the identity matrix or zero matrix, such that \( AB = BA \).
**Compute \( AD \):**
\[ AD = A \cdot D = \begin{bmatrix} 1 & 1 & 1 \\ 4 & 5 & 6 \\ 1 & 5 & 6 \end{bmatrix} \cdot \begin{bmatrix} 4 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 2 \end{bmatrix} \]
First, let's compute each element of \( AD \):
1. The element in the first row and first column of \( AD \) is: \( 1 \cdot 4 + 1 \cdot 0 + 1 \cdot 0 = 4 \).
2. The element in the first row and second column of \( AD \) is: \( 1 \cdot 0 + 1 \cdot 3 + 1 \cdot 0 = 3 \).
3. The element in the first row and third column of \( AD \) is: \( 1 \cdot 0 + 1 \cdot 0 + 1 \cdot 2 = 2 \).
4. The element in the second row and first column of \( AD \) is: \( 4 \cdot 4 + 5 \cdot 0 + 6 \cdot 0 = 16 \).
5. The element in the second row and second column of \( AD \) is: \( 4 \cdot 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0867650-b445-489d-98f3-e27e64467d14%2F85098215-c1c6-499b-a198-8a37c15b08c2%2Fg35dy9d_processed.png&w=3840&q=75)
Transcribed Image Text:**Matrix Multiplication and Properties of Diagonal Matrices**
Given matrices A and D:
\[ A = \begin{bmatrix} 1 & 1 & 1 \\ 4 & 5 & 6 \\ 1 & 5 & 6 \end{bmatrix},\quad D = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 2 \end{bmatrix} \]
We need to compute \( AD \) and \( DA \) and analyze how the columns or rows of \( A \) change when it is multiplied by \( D \) on the right or on the left. Additionally, we are to find a 3x3 matrix \( B \), not the identity matrix or zero matrix, such that \( AB = BA \).
**Compute \( AD \):**
\[ AD = A \cdot D = \begin{bmatrix} 1 & 1 & 1 \\ 4 & 5 & 6 \\ 1 & 5 & 6 \end{bmatrix} \cdot \begin{bmatrix} 4 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 2 \end{bmatrix} \]
First, let's compute each element of \( AD \):
1. The element in the first row and first column of \( AD \) is: \( 1 \cdot 4 + 1 \cdot 0 + 1 \cdot 0 = 4 \).
2. The element in the first row and second column of \( AD \) is: \( 1 \cdot 0 + 1 \cdot 3 + 1 \cdot 0 = 3 \).
3. The element in the first row and third column of \( AD \) is: \( 1 \cdot 0 + 1 \cdot 0 + 1 \cdot 2 = 2 \).
4. The element in the second row and first column of \( AD \) is: \( 4 \cdot 4 + 5 \cdot 0 + 6 \cdot 0 = 16 \).
5. The element in the second row and second column of \( AD \) is: \( 4 \cdot 0
Transcribed Image Text:### Matrix Equation Solutions
**Statement B**: There are infinitely many solutions. Any multiple of \( I_3 \) will satisfy the expression.
**Statement C**: There does not exist a matrix, \( B \), that will satisfy the expression.
#### Explanation:
- **Statement B** suggests that the solution set is infinite. Specifically, it indicates that any scalar multiple of the 3x3 identity matrix \( I_3 \) would be a valid solution to the given matrix equation.
- **Statement C** indicates that no such matrix \( B \) exists that fulfills the required condition set by the matrix equation.
These statements are typically part of a multiple-choice problem concerning the solution set of a matrix equation. Statement B reflects a situation in which a parameterization leads to an infinite number of solutions based on multiples of the identity matrix. Statement C suggests the impossibility of finding a solution. Understanding these options requires a foundational knowledge of linear algebra principles, including matrix equations, identity matrices, and solution sets.
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