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Unformatted text preview: MA 265 LECTURE NOTES: WEDNESDAY, JANUARY 30 Finding Inverses of Matrices (cont’d) Nonsingularity. Let E = E 1 E 2 ··· E k − 1 E k be a product of elementary matrices. Then E is nonsingular. We began the proof of this statement in the previous lecture. It suffices to show that each elementary matrix E i is invertible. Recall that there are three types of elementary matrices corresponding to the three types of elementary row operations. For Type I, say that we wish to interchange row i and row j . To undo this operation, we simply interchange the rows a second time: E = 1 . . . 1 1 1 . . . 1 1 1 . . . 1 = ⇒ E − 1 = 1 . . . 1 1 1 . . . 1 1 1 . . . 1 . (Recall that we constructed E by simply interchanging the i th and j th rows of the n × n identity matrix I n .) For Type II, if say that we wish to multiply row i by a nonzero number k . To undo, we simply divide row i by k : E = 1 . . . 1 k 1 . . . 1 = ⇒ E − 1 = 1 . . . 1 1 /k 1 . . . 1 . (Recall that we constructed E by multiplying the i th row of the n × n identity matrix I n by k .) For Type III, say that we wish to add a multiple of row j to row i . To undo, we simply subtract the same multiple of 1 2 MA 265 LECTURE NOTES: WEDNESDAY, JANUARY 30 row j from row i : E = 1 ....
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- Spring '10
- Linear Algebra, Invertible matrix, Identity matrix