Page 183 - 35Linear Algebra

8.3 Review Problems 183

Use row operations to put M into row echelon form. For simplicity,
2
2
1
1
1
assume that m 6= 0 6= m m − m m .
1
1
1
2
2
Prove that M is non-singular if and only if:
3
3
1
2
3
1
2
2
1
3
2
3
2
1
2
3
1
1
m m m − m m m + m m m − m m m + m m m − m m m 6= 0
2
2
3
2
3
1
1
3
2
1
2
3
2
3
1
1
1
3

0 1 a b
1
2. (a) What does the matrix E = do to M = under
2
1 0 d c
left multiplication? What about right multiplication?
1
2
(b) Find elementary matrices R (λ) and R (λ) that respectively mul-
tiply rows 1 and 2 of M by λ but otherwise leave M the same
under left multiplication.
1
(c) Find a matrix S (λ) that adds a multiple λ of row 2 to row 1
2
under left multiplication.
3. Let ˆσ denote the permutation obtained from σ by transposing the first
two outputs, i.e. ˆσ(1) = σ(2) and ˆσ(2) = σ(1). Suppose the function
f : {1, 2, 3, 4} → R. Write out explicitly the following two sums:
X X

f σ(s) and f ˆσ(s) .
σ σ
What do you observe? Now write a brief explanation why the following
equality holds
X X
F(σ) = F(ˆσ) ,
σ σ
where the domain of the function F is the set of all permutations of n
objects and ˆσ is related to σ by swapping a given pair of objects.
i
4. Let M be a matrix and S M the same matrix with rows i and j
j
switched. Explain every line of the series of equations proving that
i
det M = − det(S M).
j
0
5. Let M be the matrix obtained from M by swapping two columns i
0
and j. Show that det M = − det M.
3
6. The scalar triple product of three vectors u, v, w from R is u · (v × w).
Show that this product is the same as the determinant of the matrix
whose columns are u, v, w (in that order). What happens to the scalar
triple product when the factors are permuted?
183