Choose your favorite unique vector b in R3 with no 0 entries. What is your vector...

Choose your favorite unique vector b in R³ with no 0 entries.

What is your vector b?
Based on the “Equivalent Conditions for a Non-Singular Matrix” on page 123 in our text, why must Ax = b have a unique solution in this situation?

Solve the equation Ax = b the following three ways (you should get the same answer each way!) Recall – calculator OK, in fact suggested. (24 points – 8 points each)
- Solve the equation Ax = b using Gaussian Elimination. Show the augmented matrix and then use “ref” or “rref” on your calculator. Complete with back substitution.
- Solve the equation Ax = b using the inverse of (A), which you have already calculated above.
- Solve the equation Ax = b using Cramer’s Rule. Identify A, A_x, A_y, A_z and their determinants.
Express b as a linear combination of the column vectors of A.

12. In practice involving very large systems with square matrices of coefficients, Gaussian Elimination is often implemented by using LU factorization. Why do you think that is the case? 10-20 words (more if you want).

Replace the first row of your matrix A with the second row of your matrix A. Call this new 3x3 matrix with 2 equal rows B.

13. What is your matrix B?

14. Calculate the determinant of B without a calculator. Perform at least one row or column operation on the Determinant as part of your calculation procedure.

15. Solve the Matrix equation Bx =0

What is the general solution for x in terms of (say) any real number t?
Determine a specific non-zero solution for x (I.e. assign t to a specific real number)?
Verify that Bx = 0 for your specific solution for x.

Solutions

Expert Solution

Colby Messinger answered 1 year ago

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