Page 93 - A Course in Linear Algebra with Applications

3.2: Basic Properties of Determinants 77

2. If one row (or column) of a determinant is a scalar multiple
of another row (or column), show that the determinant is zero.
3. If A is an n x n matrix and c is a scalar, prove that
n
det(cA) = c det(A).
4. Use row operations to show that the determinant

,2 b2 „2
a c
l + a 1 + b l + c
2 2b 2 - b - 1 2c z -c-1
2a -a-1
is identically equal to zero.
5. Let A be an n x n matrix in row echelon form. Show that
det(A) equals zero if and only if the number of pivots is less
than n.

6. Use row and column operations to show that
a b c
b c a
c a b = (a + b + c)(-a 2 -b 2 - c 2 +ab + bc + ca).

Without expanding the determinant, prove that
1 1 1

x y z x - y)(y - z)(z - x)(x + y + z).
3 3
X y ^3
[Hint: show that the determinant has factors x — y , y — z ,
z — x , and that the remaining factor must be of degree 1 and
symmetric in x,y,z ].
8. Let D n denote the "bordered" n x n determinant

0 a 0 • • 0 0
b 0 a • • 0 0
0 b 0 • • 0 0

0 0 0 • • 0 a
0 0 0 • • b 0