Fundamentals of Matrix Algebra, 2011a

Chapter 3
Operaons on Matrices
1
2 R 1 → R 1
R 1 + R 2 → R 2
−3R 1 + R 3 → R 3
R 2 ↔ R 3
The first operaon mulplied a row of A by 1 2
. This means that the resulng matrix
had a determinant that was 1 2
the determinant of A.
The next two operaons did not affect the determinant at all. The last operaon,
the row swap, changed the sign. Combining these effects, we know that
−16 = det (B) = (−1) 1 det (A) .
2
Solving for det (A) we have that det (A) = 32. .
In pracce, we don’t need to keep track of operaons where we add mulples
of one row to another; they simply do not affect the determinant. Also, in pracce,
these steps are carried out by a computer, and computers don’t care about leading 1s.
Therefore, row scaling operaons are rarely used. The only things to keep track of are
row swaps, and even then all we care about are the number of row swaps. An odd
number of row swaps means that the original determinant has the opposite sign of
the triangular form matrix; an even number of row swaps means they have the same
determinant.
Let’s pracce this again.
. Example 79 The matrix B was formed from A using the following elementary
row operaons, though not necessarily in this order. Find det(A).
⎡
1 2
⎤
3
B = ⎣ 0 4 5 ⎦
0 0 6
2R 1 → R 1
1
3 R 3 → R 3
R 1 ↔ R 2
6R 1 + R 2 → R 2
S It is easy to compute det (B) = 24. In looking at our list of elementary
row operaons, we see that only the first three have an effect on the determinant.
Therefore
24 = det (B) = 2 · 1 · (−1) · det (A)
3
and hence
det (A) = −36.
.
In the previous example, we may have been tempted to “rebuild” A using the elementary
row operaons and then compung the determinant. This can be done, but
in general it is a bad idea; it takes too much work and it is too easy to make a mistake.
Let’s think some more like a mathemacian. How does the determinant work with
other matrix operaons that we know? Specifically, how does the determinant interact
with matrix addion, scalar mulplicaon, matrix mulplicaon, the transpose
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