5.7. EXERCISES 85

5.7 Exercises1. Show that matrix multiplication is associative. That is, (AB)C = A(BC) .

2. Show the inverse of a matrix, if it exists, is unique. Thus if AB = BA = I, thenB = A−1.

3. In the proof of Theorem 5.4.14 it was claimed that det(I) = 1. Here I = (δ i j) . Provethis assertion. Also prove Corollary 5.4.17.

4. Let v1, · · · ,vn be vectors in Fn and let M (v1, · · · ,vn) denote the matrix whose ith

column equals vi. Define

d (v1, · · · ,vn)≡ det(M (v1, · · · ,vn)) .

Prove that d is linear in each variable, (multilinear), that

d (v1, · · · ,vi, · · · ,v j, · · · ,vn) =−d (v1, · · · ,v j, · · · ,vi, · · · ,vn) , (5.7.22)

andd (e1, · · · ,en) = 1 (5.7.23)

where here e j is the vector in Fn which has a zero in every position except the jth

position in which it has a one.

5. Suppose f : Fn×·· ·×Fn→ F satisfies 5.7.22 and 5.7.23 and is linear in each vari-able. Show that f = d.

6. Show that if you replace a row (column) of an n× n matrix A with itself added tosome multiple of another row (column) then the new matrix has the same determinantas the original one.

7. If A = (ai j) , show det(A) = ∑(k1,··· ,kn) sgn(k1, · · · ,kn)ak11 · · ·aknn.

8. Use the result of Problem 6 to evaluate by hand the determinant

det

1 2 3 2−6 3 2 35 2 2 33 4 6 4

 .

9. Find the inverse if it exists of the matrix, et cos t sin tet −sin t cos tet −cos t −sin t

 .

10. Let Ly = y(n)+ an−1 (x)y(n−1)+ · · ·+ a1 (x)y′+ a0 (x)y where the ai are given con-tinuous functions defined on a closed interval, (a,b) and y is some function which