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