How do you factor $((1, 2, -3), (0, 1, 3), (0, 0, 1))$ into a product of elementary matrices?

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Answer 1 · 2017-01-06T20:57:08

$((1, 2, -3),(0, 1, 3),(0, 0, 1)) = ((1, 2, 0),(0, 1, 0),(0, 0, 1))((1, 0, -9),(0, 1, 0),(0, 0, 1))((1, 0, 0),(0, 1, 3),(0, 0, 1))$

Given:

$((1, 2, -3),(0, 1, 3),(0, 0, 1))$

We can describe the process of making this matrix into the identity matrix as follows:

(1) Subtract $2 xx "row2"$ from $"row1"$ to get:

$((1, 0, -9),(0, 1, 3),(0, 0, 1))$

(2) Add $9 xx "row3"$ to $"row1"$ to get:

$((1, 0, 0),(0, 1, 3),(0, 0, 1))$

(3) Subtract $3 xx "row3"$ from $"row2"$ to get:

$((1, 0, 0),(0, 1, 0),(0, 0, 1))$

Reversing the steps and expressing the row operations as elementary matrices we arrive at the following product:

$((1, 2, 0),(0, 1, 0),(0, 0, 1))((1, 0, -9),(0, 1, 0),(0, 0, 1))((1, 0, 0),(0, 1, 3),(0, 0, 1))$

How do you factor ((1, 2, -3), (0, 1, 3), (0, 0, 1))((1, 2, -3), (0, 1, 3), (0, 0, 1)) into a product of elementary matrices?