Note: In all of the following problems carry out all calculations yourself and show your work, except for the last six problems ("Maple Exercises") where you are specifically instructed to use Maple. You can use Maple to verify the correctness of your results, but giving a correct answer (likely coming out of Maple) without showing your work will receive NO credit (except for the last six problems).
1. Compute the dot product of the following two vectors. Determine if the two vectors are perpendicular. If so, explain why, if not, why not.(a)
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(b)
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2. Apply the matrix A to the vector x below (matrix-vector multiplication).
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3. Consider the following three matrices.
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Compute
(a)
| A + B |
(b)
| A · B |
(c)
| B · A |
(d)
| CT |
(e)
| ( A + B ) · CT |
4. Consider the following matrices and vectors.
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| Compute | xTATBy. |
5. Define the elementary row operations on a matrix.
6. Define the concept of an elementary matrix.
7. Are the following matrices elementary. If so, explain why, if not why not.
(a)
(b)
(c)
(d)
(e)
8. Define the concept of row-echelon form.
9. Are the following matrices in row-echelon form. If so, explain why, if not why not.
(a)
(b)
(c)
(d)
10. Define the concept of Gauss-Jordan normal form.
11. Are the following matrices in Gauss-Jordan normal form. If so, explain why, if not why not.
(a)
(b)
(c)
(d)
12. True of false. Explain why.
(a) A matrix is always row-equivalent with any of its row-echeolon forms.13. Compute the determinant of the following matrices, without Maple, showing your work.
(b) A matrix is always row-equivalent with any of Gauss-Jordan normal form.
(c) A matrix is always row-equivalent with its transpose matrix.
(d) A matrix is never row-equivalent with its transpose matrix.
(e) A matrix is always row-equivalent with its inverse matrix if it exists.
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14. Do the following matrices have an inverse. If so, explain why, if not why not.
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Maple Exercises
15. Find the row-echelon form of following matrices using Maple.| (a) |  |
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16. Find the Gauss-Jordan normal form of following matrices using Maple.
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17. Find the inverses of following matrices by their the augmented matrix, using Maple.
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18. Solve the following system of equations by using the Gauss-Jordan normal form of the augmented matrix.
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19. Solve the following system of equations by finding the inverse of the coefficient matrix.
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20. Solve the following system of equations by Cramer's Rule.
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