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Exercises prove special cases of the facts about
Chapter 2, Problem 28E(choose chapter or problem)
Problem 28E
Exercises prove special cases of the facts about elementary matrices stated in the box following Example. Here A is a 3 × 3 matrix and I = I3. (A general proof would require slightly more notation.)
Show that if row 3 of A is replaced by row3 (A) − 4 o row1 (A), the result is EA, where E is formed from I by replacing row3(I) by row3(I) - 4 o row1(I).
Example Let
Compute E1A, E2A, and E3A, and describe how these products can be obtained by elementary row operations on A.
SOLUTION Verify that
Addition of −4 times row 1 of A to row 3 produces E1A. (This is a row replacement operation.) An interchange of rows 1 and 2 of A produces E2A, and multiplication of row 3 of A by 5 produces E3A.
Questions & Answers
QUESTION:
Problem 28E
Exercises prove special cases of the facts about elementary matrices stated in the box following Example. Here A is a 3 × 3 matrix and I = I3. (A general proof would require slightly more notation.)
Show that if row 3 of A is replaced by row3 (A) − 4 o row1 (A), the result is EA, where E is formed from I by replacing row3(I) by row3(I) - 4 o row1(I).
Example Let
Compute E1A, E2A, and E3A, and describe how these products can be obtained by elementary row operations on A.
SOLUTION Verify that
Addition of −4 times row 1 of A to row 3 produces E1A. (This is a row replacement operation.) An interchange of rows 1 and 2 of A produces E2A, and multiplication of row 3 of A by 5 produces E3A.
ANSWER:
Solution
Step 1
In the question we have to show that if row 3 of A is replaced by row3 (A) − 4 . row1 (A), the result is EA, where E is formed from I by replacing row3(I) by row3(I) - 4 . row1(I).
Consider a matrix of order 3 x 3 and Identity matrix of order I = I3
