Page 20 - Matrix Analysis & Applied Linear Algebra

12 Chapter 1 Linear Equations

1.2.2. Apply Gaussian elimination with back substitution to the following sys-
tem:
2x 1 − x 2 =0,
−x 1 +2x 2 − x 3 =0,
−x 2 + x 3 =1.

1.2.3. Use Gaussian elimination with back substitution to solve the following
system:
4x 2 − 3x 3 =3,
−x 1 +7x 2 − 5x 3 =4,
−x 1 +8x 2 − 6x 3 =5.

1.2.4. Solve the following system:
x 1 + x 2 + x 3 + x 4 =1,
x 1 + x 2 +3x 3 +3x 4 =3,
x 1 + x 2 +2x 3 +3x 4 =3,
x 1 +3x 2 +3x 3 +3x 4 =4.

1.2.5. Consider the following three systems where the coefficients are the same
for each system, but the right-hand sides are different (this situation
occurs frequently):
4x − 8y +5z =1 0 0,
4x − 7y +4z =0 1 0,
3x − 4y +2z =0 0 1.
Solve all three systems at one time by performing Gaussian elimination
on an augmented matrix of the form

A b 1 b 2 b 3 .

1.2.6. Suppose that matrix B is obtained by performing a sequence of row
operations on matrix A . Explain why A can be obtained by performing
row operations on B .

1.2.7. Find angles α, β, and γ such that
2 sin α − cos β + 3 tan γ =3,
4 sin α + 2 cos β − 2 tan γ =2,
6 sin α − 3 cos β + tan γ =9,
where 0 ≤ α ≤ 2π, 0 ≤ β ≤ 2π, and 0 ≤ γ< π.