Page 102 - A Course in Linear Algebra with Applications

86 Chapter Three: Determinants

verify the identity det(AB) = det(A) det(B).
2. By finding the relevant adjoints, compute the inverses of
the following matrices:

4
(a) (b) (c)
-2

3. If A is a square matrix and n is a positive integer, prove
n n
that det(A ) = (det(A)) .
4. Use Cramer's Rule to solve the following linear systems:

2x x - 3x 2 + £3 = -1
(a) Xi + 3x 2 + x 3 = 6
2xi + x 2 + x 3 = 11

Xi + x 2 + x 3 = -1
(b) 2xi - x 2 - x 3 = 4
Xi + 2x 2 - 3x 3 = 7
5. Let A be an n x n matrix. Prove that A is invertible if and
only if adj(A) is invertible.
6. Let A be any n x n matrix where n > 1. Prove that
det(adj(yl)) = (det(A)) n_1 . [Hint: first deal with the case
where det(A) ^ 0, by applying det to each side of the identity
of 3.3.6. Then argue that the result must still be true when
det(A)=0].

7. Find the equation of the plane which contains the points
(1,1,-2), 1,-2, 7) and 0,1,-4).
(
(
8. Consider the four points in three dimensional space
Pi(xi,yi,Zi), i = 1, 2, 3, 4. Prove that a necessary and suffi-
cient condition for the four points to lie in a plane is
xi y\ zi 1

X 2 V2 Z 2 1
= 0.
X3 J/3 Z3 1
1
x 4 y 4 z A