4. Find the inverse of D, D¹¹, by three different methods: a. entering D in your TI-84 calculator and raising it to the negative one power (the "x¹" button works). b. reducing the matrix [D I] to reduced echelon form. c. using the formula D-¹ = 1 adj D. Show the steps of det D finding the adjugate and the determinant. Verify that the inverses you found are all identical. If they are not, go back and fix them. 5. Show that the inverse you found is correct by calculating DD¹¹ and D¹D. What result should you get? Did you get that? 6. Solve the equation Dx = f by three different methods: a. writing the augmented matrix and reducing it to echelon form. left-multiplying by D¹¹ on both b. sides.
4. Find the inverse of D, D¹¹, by three different methods: a. entering D in your TI-84 calculator and raising it to the negative one power (the "x¹" button works). b. reducing the matrix [D I] to reduced echelon form. c. using the formula D-¹ = 1 adj D. Show the steps of det D finding the adjugate and the determinant. Verify that the inverses you found are all identical. If they are not, go back and fix them. 5. Show that the inverse you found is correct by calculating DD¹¹ and D¹D. What result should you get? Did you get that? 6. Solve the equation Dx = f by three different methods: a. writing the augmented matrix and reducing it to echelon form. left-multiplying by D¹¹ on both b. sides.
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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They are all linked. Could you please provide me with an answer for all three numbers?
![### Matrix Inversion and Solution Verification
Consider the matrix \( D \) and vector \( f \) given by:
\[
D = \begin{bmatrix} 5 & 0 & 0 \\ -8 & 1 & -4 \\ 3 & -6 & 0 \end{bmatrix}, \quad f = \begin{bmatrix} -1 \\ 1 \\ 3 \end{bmatrix}
\]
#### 4. Find the inverse of \( D \), \( D^{-1} \), by three different methods:
a. **Using a TI-84 Calculator:**
- Enter matrix \( D \) into your calculator.
- Use the "x\( ^{-1} \)" button to compute \( D^{-1} \).
b. **Row Reduction:**
- Reduce the matrix \([D \, | \, I]\) to its reduced row echelon form to find \( D^{-1} \).
c. **Formula Method:**
- Use the formula \( D^{-1} = \frac{1}{\det D} \text{adj} D \).
- Show the steps of finding the adjugate and the determinant of \( D \).
Verify that the inverses you found are all identical. If they are not, go back and fix them.
#### 5. Verify the Inverse:
- Check the correctness of your inverse by calculating \( DD^{-1} \) and \( D^{-1}D \).
- Confirm both results are the identity matrix. Did you achieve this result?
#### 6. Solve the Equation \( Dx = f \) by Three Methods:
a. **Augmented Matrix Reduction:**
- Write the augmented matrix and reduce it to echelon form.
b. **Using the Inverse:**
- Left-multiply both sides by \( D^{-1} \) to solve for \( x \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F44c94bd2-8c1f-4d24-8d96-8ec15a0486f1%2F45570bc4-e305-44ce-8423-4897e888bb63%2Fmltllhl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Matrix Inversion and Solution Verification
Consider the matrix \( D \) and vector \( f \) given by:
\[
D = \begin{bmatrix} 5 & 0 & 0 \\ -8 & 1 & -4 \\ 3 & -6 & 0 \end{bmatrix}, \quad f = \begin{bmatrix} -1 \\ 1 \\ 3 \end{bmatrix}
\]
#### 4. Find the inverse of \( D \), \( D^{-1} \), by three different methods:
a. **Using a TI-84 Calculator:**
- Enter matrix \( D \) into your calculator.
- Use the "x\( ^{-1} \)" button to compute \( D^{-1} \).
b. **Row Reduction:**
- Reduce the matrix \([D \, | \, I]\) to its reduced row echelon form to find \( D^{-1} \).
c. **Formula Method:**
- Use the formula \( D^{-1} = \frac{1}{\det D} \text{adj} D \).
- Show the steps of finding the adjugate and the determinant of \( D \).
Verify that the inverses you found are all identical. If they are not, go back and fix them.
#### 5. Verify the Inverse:
- Check the correctness of your inverse by calculating \( DD^{-1} \) and \( D^{-1}D \).
- Confirm both results are the identity matrix. Did you achieve this result?
#### 6. Solve the Equation \( Dx = f \) by Three Methods:
a. **Augmented Matrix Reduction:**
- Write the augmented matrix and reduce it to echelon form.
b. **Using the Inverse:**
- Left-multiply both sides by \( D^{-1} \) to solve for \( x \).
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