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hw4-solutions

# hw4-solutions - Assignment 4 Determinants Cramer's Rule...

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Assignment 4: Determinants, Cramer's Rule, Vector Spaces Written by: Saber Mirzaei Problem 1. (a) | | → | | → | | Based on the properties of determinant, if two rows or two columns of a matrix are equal then the determinant is zero. (b) | | → | | | | If all elements of a row or a column of a matrix are zero then the determinant is zero. (c) | | | | | | | | | | Based on the properties of determinant, if two rows or two columns of a matrix are equal then the determinant is zero.

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Problem 2. (a) Assume | | → | | → | | → | | → | | | | (b) . For the value of det is zero which means . You can check the truth by plugging in the values and using Matlab/Octave.
Problem 3. a. Row operations don’t change the determinant of a matrix. The resulting matrix is: | | b. The resulting matrix is: | | c. The resulting matrix is lower triangular hence:

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Problem 4. (a) The corresponding matrix equation is: [ ] [ ] [ ] This system has a unique solution iff . Hence: | | which always holds. Using Cramer’s Rule: and . So (b) The corresponding matrix equation is: [ ] [ ] [ ] This system has a unique solution iff . Hence: | |
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