Prove that A is idempotent if and only if A' is idempotent. Getting Started: The phrase "if and only if" means that you have to prove two statemen 1. If A is idempotent, then A' is idempotent. 2. If AT is idempotent, then A is idempotent. STEP 1: Begin your proof of the first statement by assuming that A is idempotent. This means that A STEP 2: Take the transpose of both sides of the equation from Step 1. T STEP 3: Use the properties of the transpose to simplify your result from Step 2. T This shows that A' is idempotent. STEP 4: Begin your proof of the second statement by assuming that A is idempotent. This means that A' = STEP 5: Take the transpose of both sides of the equation from Step 4. STEP 6: Use the properties of the transpose to simplify your result from Step 5. %3D This shows that A is idempotent.
Prove that A is idempotent if and only if A' is idempotent. Getting Started: The phrase "if and only if" means that you have to prove two statemen 1. If A is idempotent, then A' is idempotent. 2. If AT is idempotent, then A is idempotent. STEP 1: Begin your proof of the first statement by assuming that A is idempotent. This means that A STEP 2: Take the transpose of both sides of the equation from Step 1. T STEP 3: Use the properties of the transpose to simplify your result from Step 2. T This shows that A' is idempotent. STEP 4: Begin your proof of the second statement by assuming that A is idempotent. This means that A' = STEP 5: Take the transpose of both sides of the equation from Step 4. STEP 6: Use the properties of the transpose to simplify your result from Step 5. %3D This shows that A is idempotent.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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![## Prove that \( A \) is idempotent if and only if \( A^T \) is idempotent.
### Getting Started:
The phrase "if and only if" means that you have to prove two statements.
1. If \( A \) is idempotent, then \( A^T \) is idempotent.
2. If \( A^T \) is idempotent, then \( A \) is idempotent.
### STEP 1:
Begin your proof of the first statement by assuming that \( A \) is idempotent. This means that
\[ A^2 = \boxed{A}. \]
### STEP 2:
Take the transpose of both sides of the equation from Step 1.
\[ (A^2)^T = A^T. \]
### STEP 3:
Use the properties of the transpose to simplify your result from Step 2.
\[ (A \cdot A)^T = A^T \]
\[ A^T \cdot (A^T)^T = A^T \]
\[ A^T \cdot A^T = A^T \]
This shows that \( A^T \) is idempotent.
### STEP 4:
Begin your proof of the second statement by assuming that \( A^T \) is idempotent. This means that
\[ (A^T)^2 = \boxed{A^T}. \]
### STEP 5:
Take the transpose of both sides of the equation from Step 4.
\[ [(A^T)^2]^T = (A^T)^T. \]
### STEP 6:
Use the properties of the transpose to simplify your result from Step 5.
\[ [A^T \cdot A^T]^T = (A^T)^T \]
\[ (A^T)^T \cdot A^T = A \]
\[ A \cdot A = A \]
This shows that \( A \) is idempotent.
By completing these steps, we have shown that \( A \) is idempotent if and only if \( A^T \) is idempotent.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F73fb2883-df73-4e37-b7ef-66d24e1d1095%2Fc31a7859-754b-443b-b00b-75d4d8141a0a%2Fcgvhc3r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Prove that \( A \) is idempotent if and only if \( A^T \) is idempotent.
### Getting Started:
The phrase "if and only if" means that you have to prove two statements.
1. If \( A \) is idempotent, then \( A^T \) is idempotent.
2. If \( A^T \) is idempotent, then \( A \) is idempotent.
### STEP 1:
Begin your proof of the first statement by assuming that \( A \) is idempotent. This means that
\[ A^2 = \boxed{A}. \]
### STEP 2:
Take the transpose of both sides of the equation from Step 1.
\[ (A^2)^T = A^T. \]
### STEP 3:
Use the properties of the transpose to simplify your result from Step 2.
\[ (A \cdot A)^T = A^T \]
\[ A^T \cdot (A^T)^T = A^T \]
\[ A^T \cdot A^T = A^T \]
This shows that \( A^T \) is idempotent.
### STEP 4:
Begin your proof of the second statement by assuming that \( A^T \) is idempotent. This means that
\[ (A^T)^2 = \boxed{A^T}. \]
### STEP 5:
Take the transpose of both sides of the equation from Step 4.
\[ [(A^T)^2]^T = (A^T)^T. \]
### STEP 6:
Use the properties of the transpose to simplify your result from Step 5.
\[ [A^T \cdot A^T]^T = (A^T)^T \]
\[ (A^T)^T \cdot A^T = A \]
\[ A \cdot A = A \]
This shows that \( A \) is idempotent.
By completing these steps, we have shown that \( A \) is idempotent if and only if \( A^T \) is idempotent.
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