Evaluate the determinants given in Exercises 21–23.
21.
7 0 −1 0
2 0 1 3
1 −3 0 2
0 5 1 −2
22.
8 0 0 0
15 1 0 0
−7 6 −1 0
8 1 9 7
23.
5 −1 0 8 11
0 2 1 9 7
0 0 4 −3 5
0 0 0 2 1
0 0 0 0 −3
24. Prove that a matrix that has a row or a column con- sisting entirely of zeros has determinant equal to zero.
25. Anupper triangularmatrix is ann×nmatrix whose entries below the main diagonal are all zero. (Note:
Themain diagonalis the diagonal going from upper left to lower right.) For example, the matrix
⎡
⎢⎣
1 2 −1 2
0 3 4 3
0 0 5 6
0 0 0 7
⎤
⎥⎦
is upper triangular.
(a) Give an analogous definition for alower triangu- larmatrix and also an example of one.
(b) Use cofactor expansion to show that the determi- nant of anyn×nupper or lower triangular matrix Ais the product of the entries on the main diagonal.
That is, detA=a11a22· · ·ann.
26. Some properties of the determinant. Exercises 24 and 25 show that it is not difficult to compute de- terminants of even large matrices, provided that the matrices have a nice form. The following operations (calledelementary row operations) can be used to transform ann×nmatrix into one in upper triangular form:
I. Exchange rowsiandj.
II. Multiply rowiby a nonzero scalar.
III. Add a multiple of rowito row j. (Rowiremains unchanged.)
For example, one can transform the matrix
⎡
⎣ 0 2 3
1 7 −2
1 5 9
⎤
⎦
into one in upper triangular form in three steps:
Step 1. Exchange rows 1 and 2 (this puts a nonzero entry in the upper left corner):
⎡
⎣ 0 2 3
1 7 −2
1 5 9
⎤
⎦ −→
⎡
⎣ 1 7 −2
0 2 3
1 5 9
⎤
⎦. Step 2. Add−1 times row 1 to row 3 (this eliminates the nonzero entries below the entry in the upper left corner):
⎡
⎣ 1 7 −2
0 2 3
1 5 9
⎤
⎦ −→
⎡
⎣ 1 7 −2
0 2 3
0 −2 11
⎤
⎦. Step 3. Add row 2 to row 3:
⎡
⎣ 1 7 −2
0 2 3
0 −2 11
⎤
⎦ −→
⎡
⎣ 1 7 −2
0 2 3
0 0 14
⎤
⎦. The question is, how do these operations affect the de- terminant?
(a) By means of examples, make a conjecture as to the effect of a row operation of type I on the determi- nant. (That is, if matrixBresults from matrixAby performing a single row operation of type I, how are detAand detBrelated?) You need not prove your results are correct.
(b) Repeat part (a) in the case of a row operation of type III.
(c) Prove that ifB results from Aby multiplying the entries in theith row ofAby the scalarc(a type II operation), then detB=c·det A.
27. Calculate the determinant of the matrix
A=
⎡
⎢⎢
⎢⎣
2 1 −2 7 8
1 0 1 −2 4
−1 1 2 3 −5
0 2 3 1 7
−3 2 −1 0 1
⎤
⎥⎥
⎥⎦
by using row operations to transform Ainto a matrix in upper triangular form and by using the results of Exercise 26 to keep track of how the determinant ofA and the determinant of your final matrix are related.
28. (a) Is det(A+B)=detA+detB? Why or why not?
(b) Calculate
1 2 7
3+2 1−1 5+1
0 −2 0
and
1 2 7
3 1 5
0 −2 0
+
1 2 7
2 −1 1
0 −2 0
, and compare your results.
(c) Calculate
1 3 2+3
0 4 −1+5
−1 0 0−2 and
1 3 2
0 4 −1
−1 0 0 +
1 3 3
0 4 5
−1 0 −2 , and compare your results.
(d) Conjecture and prove a result about sums of deter- minants. (You may wish to construct further exam- ples such as those in parts (b) and (c).)
29. It is a fact that, ifAandBare anyn×nmatrices, then det(A B)=(detA)(detB).
Use this fact to show that det(A B)=det(B A). (Recall that A B=B A, in general.)
Ann×nmatrixAis said to beinvertible(ornonsingular) if there is anothern×nmatrixBwith the property that
A B=B A=In,
whereIndenotes then×nidentity matrix. (See Exercise 20.) The matrixBis called aninverseto the matrix A. Exercises 30–38 concern various aspects of matrices and their inverses.
30. (a) Verify that 1 0
1 1
is an inverse of
1 0
−1 1
.
(b) Verify that
⎡
⎣ 1 2 3 2 5 3 1 0 8
⎤
⎦ is an inverse of
⎡
⎣ −40 16 9 13 −5 −3
5 −2 −1
⎤
⎦.
31. Using the definition of an inverse matrix, find an inverse to
⎡
⎣ 2 2 1
0 1 0
0 0 −1
⎤
⎦.
32. Try to find an inverse matrix to
⎡
⎣ 0 2 1
0 1 0
0 0 −1
⎤
⎦. What happens?
33. Show that if ann×nmatrixAis invertible, thenAcan have only one inverse matrix. Thus, we may writeA−1 to denote the unique inverse of a nonsingular matrix A. (Hint: SupposeAwere to have two inverses Band C. ConsiderB(AC).)
34. Suppose that AandBare n×ninvertible matrices.
Show that the product matrixA Bis invertible by ver- ifying that its inverse (A B)−1= B−1A−1.
35. (a) Show that if A is invertible, then detA=0. (In fact, the converse is also true.)
(b) Show that if A is invertible, then det(A−1)= 1
detA.
36. (a) Show that, ifad−bc=0, then a general 2×2 matrix
a b c d
has the matrix 1
ad−bc
d −b
−c a
= d
ad−bc −ad−bbc
−ad−bcc ad−bca
as inverse.
(b) Use this formula to find an inverse of
2 4
−1 2
. 37. If A is a 3×3 matrix and detA=0, then there is a (somewhat complicated) formula for A−1. In particular,
A−1= 1 detA
⎡
⎣ |A11| −|A21| |A31|
−|A12| |A22| −|A32|
|A13| −|A23| |A33|
⎤
⎦,
where Ai j denotes the submatrix of A obtained by deleting the ith row and jth column (see Defini- tion 6.5). Use this formula to find the inverse of
A=
⎡
⎣ 2 1 1 0 2 4 1 0 3
⎤
⎦.
More generally, ifAis anyn×nmatrix and detA=0, then
A−1= 1 detAadjA,
where adjAis theadjoint matrixof A, that is, the matrix whosei jth entry is (−1)i+j|Aj i|. (Note: The formula for the inverse matrix using the adjoint is typi- cally more of theoretical than practical interest, as there are more efficient computational methods to determine the inverse, when it exists.)
38. Repeat Exercise 37 with the matrix
A=
⎡
⎣ 2 −1 3
1 2 −2
3 0 1
⎤
⎦.
Cross products in Rn.Although it is not possible to define a cross product of two vectors inRn as we did for two vectors inR3, we can construct a “cross product” of n−1vectors inRnthat behaves analogously to the three-dimensional cross
product. To be specific, if
a1=(a11,a12, . . . ,a1n), a2=(a21,a22, . . . ,a2n), . . . , an−1=(an−11,an−12, . . . ,an−1n)
aren−1vectors inRn, we definea1×a2×· · ·×an−1to be the vector inRngiven by the symbolic determinant
a1×a2×· · ·×an−1=
e1 e2 · · · en
a11 a12 · · · a1n
a21 a22 · · · a2n
... ... . .. ... an−1 1 an−1 2 · · · an−1n
.
(Heree1, . . . ,enare the standard basis vectors forRn.)Exer- cises 39–42 concern this generalized notion of cross product.
39. Calculate the following cross product inR4: (1,2,−1,3)×(0,2,−3,1)×(−5,1,6,0).
40. Use the results of Exercises 26 and 28 to show that (a) a1× · · · ×ai× · · · ×aj× · · · ×an−1
= −(a1× · · · ×aj× · · · ×ai× · · · ×an−1), 1≤i≤n−1,1≤ j ≤n−1
(b) a1× · · · ×kai× · · · ×an−1
=k(a1× · · · ×ai× · · · ×an−1), 1≤i≤n−1.
(c) a1× · · · ×(ai+b)× · · · ×an−1
=a1× · · · ×ai× · · · ×an−1+ a1× · · · × b× · · · ×an−1, 1≤i≤n−1, allb∈Rn.
(d) Show that ifb=(b1, . . . ,bn) is any vector inRn, then
b·(a1×a2×· · ·×an−1) is given by the determinant
b1 · · · bn
a11 · · · a1n
... ...
an−11 · · · an−1n
.
41. Show that the vector b=a1×a2×· · ·×an−1 is orthogonal toa1, . . . ,an−1.
42. Use the generalized notion of cross products to find an equation of the (four-dimensional) hyper- plane inR5 through the five points P0(1,0,3,0,4), P1(2,−1,0,0,5), P2(7,0,0,2,0), P3(2,0,3,0,4), andP4(1,−1,3,0,4).