Page 62 - 35Linear Algebra
P. 62
62 Systems of Linear Equations
1. While performing Gaussian elimination on these augmented matrices
write the full system of equations describing the new rows in terms of
the old rows above each equivalence symbol as in Example 21.
1 1 0 5
2 2 10
, 1 1 −1 11
1 2 8
−1 1 1 −5
2. Solve the vector equation by applying ERO matrices to each side of
the equation to perform elimination. Show each matrix explicitly as in
Example 24.
3 6 2 x −3
5 9 4 y = 1
2 4 2 z 0
3. Solve this vector equation by finding the inverse of the matrix through
(M|I) ∼ (I|M −1 ) and then applying M −1 to both sides of the equation.
2 1 1 x 9
1 1 1 y = 6
1 1 2 z 7
4. Follow the method of Examples 29 and 30 to find the LU and LDU
factorization of
3 3 6
3 5 2 .
6 2 5
5. Multiple matrix equations with the same matrix can be solved simul-
taneously.
(a) Solve both systems by performing elimination on just one aug-
mented matrix.
2 −1 −1 x 0 2 −1 −1 a 2
−1 1 1 y = 1 , −1 1 1 b = 1
1 −1 0 z 0 1 −1 0 c 1
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