Algebra Know-It-ALL

304 The Matrix Morphing Game

Let’s make the number 133 vanish. We can use the third row as the “weapon” here. If we

multiply the third row by 133, we get

3 − 2 08

0 190 133 209

00 − 133 − 399

Adding the second two rows and then replacing the second row with the sum, we obtain

3 − 2 08

0 190 0 − 190

(^00) − 133 − 399
We now have an opportunity to reduce the sizes of the numbers in the second and third rows.
Let’s divide the second row through by 190, and divide the third row through by −133. That
gives us
3 − 2 08
(^010) − 1
0013
We have only to make the element −2 in the first row vanish, and we’ll have the matrix in
diagonal form. We can multiply the second row by 2 to obtain
3 − 2 08
020 − 2
0013
Adding the first and second rows and then replacing the first row with the sum, we get
3006
020 − 2
0013
Deriving the unit diagonal form
Our remaining task is simple and clean. We can divide the top row by 3 and the middle row
by 2, obtaining the unit diagonal matrix
1002
010 − 1
0013