Page 178 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

is , . is invertible and

Prove: If A is invertible, then
7.

Prove: If A is an matrix, then .
8.

9. (For Readers Who Have Studied Calculus) Show that if , , , and are differentiable functions,
and if

10. can be expressed as
(a) In the accompanying figure, the area of the triangle

Use this and the fact that the area of a trapezoid equals the altitude times the sum of
the parallel sides to show that

Note In the derivation of this formula, the vertices are labeled such that the triangle is
traced counterclockwise proceeding from to to . For a clockwise

orientation, the determinant above yields the negative of the area.

(b) Use the result in (a) to find the area of the triangle with vertices (3, 3), (4, 0), .

Figure Ex-10

Prove: If the entries in each row of an matrix A add up to zero, then the determinant of A is zero.
11.

Hint Consider the product , where is the matrix, each of whose entries is one.