Choose your favorite unique vectorb in R³ with no 0 entries.

8. What is your vector b?
9. 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?

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

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

Replace the first row of your matrix A with the second row ofyour 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 acalculator. Perform at least one row or columnoperation on the Determinant as part of your calculationprocedure.

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 toa specific real number)?
• Verify that Bx = 0 for yourspecific solution for x.