Linear Algebra, 2020a

Section IV. Matrix Operations 251
and then add −1 times the first row to the second.
⎛ ⎞ ⎛
1 0 0
⎜ ⎟ ⎜
1 6 0 9
⎞ ⎛
⎟ ⎜
1 6 0 9
⎞
⎟
⎝−1 1 0⎠
⎝1 5 −2 2⎠ = ⎝0 −1 −2 −7⎠
0 0 1 0 0 3 9 0 0 3 9
Now back substitution will give the solution.
3.22 Example Gauss-Jordan reduction works the same way. For the matrix ending
the prior example, first turn the leading entries to ones,
⎛
⎜
1 0 0
⎞ ⎛
⎞ ⎛
1 6 0 9
⎟ ⎜
⎟ ⎜
1 6 0 9
⎞
⎟
⎝0 −1 0 ⎠ ⎝0 −1 −2 −7⎠ = ⎝0 1 2 7⎠
0 0 1/3 0 0 3 9 0 0 1 3
then clear the third column, and then the second column.
⎛
⎜
1 −6 0
⎞ ⎛
⎟ ⎜
1 0 0
⎞ ⎛
⎟ ⎜
1 6 0 9
⎞ ⎛
⎟ ⎜
1 0 0 3
⎞
⎟
⎝0 1 0⎠
⎝0 1 −2⎠
⎝0 1 2 7⎠ = ⎝0 1 0 1⎠
0 0 1 0 0 1 0 0 1 3 0 0 1 3
3.23 Corollary For any matrix H there are elementary reduction matrices R 1 , ...,
R r such that R r · R r−1 ···R 1 · H is in reduced echelon form.
Until now we have taken the point of view that our primary objects of study
are vector spaces and the maps between them, and we seemed to have adopted
matrices only for computational convenience. This subsection shows that this
isn’t the entire story.
Understanding matrix operations by understanding the mechanics of how
the entries combine is also useful. In the rest of this book we shall continue to
focus on maps as the primary objects but we will be pragmatic — if the matrix
point of view gives some clearer idea then we will go with it.
Exercises
̌ 3.24 Predict the result of each product with a permutation matrix and then check
by multiplying it out.
⎛ ⎞ ⎛ ⎞
( )( ) ( )( ) 1 0 0 1 2 3
0 1 1 2 1 2 0 1
(a)
(b)
(c) ⎝0 0 1⎠
⎝4 5 6⎠
1 0 3 4 3 4 1 0
0 1 0 7 8 9
̌ 3.25 Predict the result of each multiplication by an elementary reduction matrix,
and then
(
check
)(
by multiplying
) (
it out.
)( ) ( )( )
3 0 1 2 1 0 1 2
1 0 1 2
(a)
(b)
(c)
0 1 3 4 0 2 3 4 −2 1 3 4
( )( ) ( )( )
1 2 1 −1 1 2 0 1
(d)
(e)
3 4 0 1
3 4 1 0
3.26 Predict the result of each multiplication by a diagonal matrix, and then check
by multiplying it out.