2014 Winter Homework 5 Solutions - Linear Algebra(MAT-022A Solution to writing assignment 5 Section 2.5 24 Use Gauss-Jordan elimination on[U I to nd the

# 2014 Winter Homework 5 Solutions - Linear Algebra(MAT-022A...

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Linear Algebra (MAT-022A) Solution to writing assignment 5 Section 2.5 24. Use Gauss-Jordan elimination on [ U I ] to find the upper triangular U - 1 : UU - 1 = I 1 a b 0 1 c 0 0 1 x 1 x 2 x 3 = 1 0 0 0 1 0 0 0 1 . Solution. Use Gauss-Jordan elimination to obtain 1 a b 1 0 0 0 1 c 0 1 0 0 0 1 0 0 1 -→ 1 a 0 1 0 - b 0 1 0 0 1 - c 0 0 1 0 0 1 (replace row 1 by (row 1) - b × (row 3)) (replace row 2 by (row 2) - c × (row 3)) -→ 1 0 0 1 - a ac - b 0 1 0 0 1 - c 0 0 1 0 0 1 (replace row 1 by (row 1) - a × (row 2)) , from which we have U - 1 = 1 - a ac - b 0 1 - c 0 0 1 . 40. A is a 4 by 4 matrix with 1’s on the diagonal and - a, - b, - c on the diagonal above. Find A - 1 for this bidiagonal matrix. 1
Solution. Use Gauss-Jordan elimination to obtain 1 - a 0 0 1 0 0 0 0 1 - b 0 0 1 0 0 0 0 1 - c 0 0 1 0 0 0 0 1 0 0 0 1 -→ 1 - a 0 0 1 0 0 0 0 1 - b 0 0 1 0 0 0 0 1 0 0 0 1 c 0 0 0 1 0 0 0 1 (replace row 3 by (row 3)+ c × (row 4)) -→ 1 - a 0 0 1 0 0 0 0 1 0 0 0 1 b bc 0 0 1 0 0 0 1 c 0 0 0 1 0 0 0 1 (replace row 3 by (row 2)+ b × (row 3)) -→ 1 0 0 0 1 a ab abc 0 1 0 0 0 1 b bc 0 0 1 0 0 0 1 c 0 0 0 1 0 0 0 1