In Exercises 75-78. write the indicated colwnn of A=[~ -2

For each ofrhe mmrices A and Bin Exercises 1- 8, derennine whether B = A 1 . A= [: 8 = [: 0 -I 0 -n 0.5] and -

For each ofrhe mmrices A and Bin Exercises 1- 8, derennine whether B = A 1 A=[~ ;

For each ofrhe mmrices A and Bin Exercises 1- 8, derennine whether B = A 1 A= [i

For each ofrhe mmrices A and Bin Exercises 1- 8, derennine whether B = A 1 A= ~ I n

For each ofrhe mmrices A and Bin Exercises 1- 8, derennine whether B = A 1 A" [ i

For each ofrhe mmrices A and Bin Exercises 1- 8, derennine whether B = A 1 ~ [j 2 2 ll 6. I I and 3

For each ofrhe mmrices A and Bin Exercises 1- 8, derennine whether B = A 1 ~ [j 2 2 ll 6. I I and 3

For each ofrhe mmrices A and Bin Exercises 1- 8, derennine whether B = A 1 ~ [j 2 2 ll 6. I I and 3

For Exercises 9- 14, find file value of each malrix expression. where A and 8 are life inverlible 3 x 3 mal rices sucil1ila1 A- = n 2 _:] ~ - I l AT)-1

For Exercises 9- 14, find file value of each malrix expression. where A and 8 are life inverlible 3 x 3 mal rices sucil1ila1 A- = n 2 _:] ~ - I l (BT)-l

For Exercises 9- 14, find file value of each malrix expression. where A and 8 are life inverlible 3 x 3 mal rices sucil1ila1 A- = n 2 _:] ~ - I l B)-1

For Exercises 9- 14, find file value of each malrix expression. where A and 8 are life inverlible 3 x 3 mal rices sucil1ila1 A- = n 2 _:] ~ - I l (BA)-1

For Exercises 9- 14, find file value of each malrix expression. where A and 8 are life inverlible 3 x 3 mal rices sucil1ila1 A- = n 2 _:] ~ - I l (AB'I)-1

For Exercises 9- 14, find file value of each malrix expression. where A and 8 are life inverlible 3 x 3 mal rices sucil1ila1 A- = n 2 _:] ~ - I l TB1)-l

In Exercises 15- 22, find file inverse of each elemelllaty malrix. :

In Exercises 15- 22, find file inverse of each elemelllaty malrix. 1 -3

In Exercises 15- 22, find file inverse of each elemelllaty malrix. -i 0 ]

In Exercises 15- 22, find file inverse of each elemelllaty malrix. 0101001

In Exercises 15- 22, find file inverse of each elemelllaty malrix. 100000

In Exercises 15- 22, find file inverse of each elemelllaty malrix. 0100

In Exercises 15- 22, find file inverse of each elemelllaty malrix.1000000

In Exercises 15- 22, find file inverse of each elemelllaty malrix.1000000

In Exercises 23- 32, find tm elementary nwtri.x E such film EA = B [~ ~] and [-1 8 = 3 -~

In Exercises 23- 32, find tm elementary nwtri.x E such film EA = B A= [-1 2 ~] and [- 1 8

In Exercises 23- 32, find tm elementary nwtri.x E such film EA = B =[; - 3] 10 and 8 = [

In Exercises 23- 32, find tm elementary nwtri.x E such film EA = B =[; - 3] 10 and 8 = [

In Exercises 23- 32, find tm elementary nwtri.x E such film EA = B = [ ~ 2 -

In Exercises 23- 32, find tm elementary nwtri.x E such film EA = B 2 I ] 8 = [i - 5 28.

In Exercises 23- 32, find tm elementary nwtri.x E such film EA = B i

In Exercises 23- 32, find tm elementary nwtri.x E such film EA = B A= [ ~ 2 ~] 8 = ~ 2 -

In Exercises 23- 32, find tm elementary nwtri.x E such film EA = B [' 2 ] ~ 31. A= 4 5 a

In Exercises 23- 32, find tm elementary nwtri.x E such film EA = B = [ -l 2 3 !] I 3 and -I 0 8 = [ -~ 2 3 j] I

Every square matrix is invenible.

Invertible matrices arc square.

Elementary matrices are invertible.

If A and 8 are matrices such that AB = !,. for some 11, then both A and 8 arc invertible

If 8 and C are inverses of a matrix A, then 8 =C.

f A and 8 are invenible n x 11 matrices, then AB T is invertible.

An invertible matrix may have more than one inverse.

For any matrices A and 8, if A is the inverse of 8 , then 8 is the inverse of A.

For any matrice.~ A and 8 , if A is the inverse of Br. then A is the transpose of 8 -

If A and 8 are invertible 11 x 11 matrices, then AB is also inve1tible.

If A and 8 are invertible 11 x n matrices, then (AB)- 1 = A- 18 - 1

An e lementary matrix is a matrix that can be obtai ned by a sequence of elementary row operations on an identity matrix.

An elementary 11 x 11 matrix has at most 11 + I nonzero entries

An elementary 11 x 11 matrix has at most 11 + I nonzero entries

Every elementary matrix is invertible.

If A and 8 are m x 11 matrices and 8 can be obtained from A by an e lememary row operation on A, then there is an elementary m x m matrix E such that 8 = EA.

If R is the reduced row echelon form of a matrix A, then there exists an invertible matrix P s uch that PA = R.

Let R be the reduced row echelon form of a matrix A. If colunm j of R is a linear combination of the previous columns of R, then column j of A is a linear combination of the previous columns of A.

The pivot columns of a matrix are linearly dependent.

Every column of a matrL' is a linear combination of its pivot columns

3 Let A., be the a-rotation matrix, Prove that (A,)T = (A,)-1

Let A = [: ~ l (a) Suppose ad - be # 0. and 8 ---I [ d - ad- be -c Show that A8 = 8A = /, and hence A and B =A-1. - is invertible (b) Prove the converse of (a): If A is invertible, then tul- be# 0

Prove that the product of e lementary matrices is invertible

(a) Let A be an invertible 11 x 11 matrix, and let u and v be vectors in n" such that u # v. Prove that Au # Av, (b) Find a 2 x 2 matrix A and distinct vectors u and v in n2 such that Au = Av.

Let Q be an invertible 11. x 11 matrix, Prove that the subset {u1, u 2, ... , uk} of n" is linearly independent if and only if {Q ~o Q u2, .. . , Qud is linearly independent.

Prove Theorem 2.2(a).

Prove that if A, 8, and C are invertible 11 x 11 matrices, then ABC is invettible and (ABC)- 1 = c - n - tA- 1.

Let A and 8 be n x 11 matrices such that both A and AB are invertible. Prove that 8 is invettible, by writing it as the product of two invertible matrices,

Let A and 8 be 11 x 11 matrices such that AB = 1,.. Prove that the rank of A is 11. Him: Theorem 1.6 can be useful.

Prove that if A is an 111 x 11 matrix and 8 is an 11 x p matrix, then rank A8 ::; rank 8. Hi11t: Prove that if the kth column of 8 is not a pivot column of 8. then the kth column of AB is not a pivot column of AB.

Prove that if 8 is an 11 x 11 matrix with rank 11 , then there exists an 11 x 11 matrix C such that BC = 1,.. Him: Theorem 1.6 can be useful.

Prove that if A and 8 arc 11 x 11 matrices such that A8 = 1,., then B is invertible and A= s - t. Him: Use Exercises 62 and 63.

Prove that if an 11 x 11 matrix has rank 11, then it is invertible. Him: Use Exercises 63 and 64.

Let M = 02 8 , where A and 8 are square and Ot ;md 0 2 arc zero matrices. Prove that M is invct1iblc if and only if A and 8 are both invertible. Him: Think about the reduced row echelon form of M.

/11 Exercises 67- 74, find the matrix A, given the retluced row echelon jon11 R of A a11d i11jormatio11 abow cenai11 co/wm1s of A. R = [~

/11 Exercises 67- 74, find the matrix A, given the retluced row echelon jon11 R of A a11d i11jormatio11 abow cenai11 co/wm1s of A. 12000

/11 Exercises 67- 74, find the matrix A, given the retluced row echelon jon11 R of A a11d i11jormatio11 abow cenai11 co/wm1s of A. R = [~

/11 Exercises 67- 74, find the matrix A, given the retluced row echelon jon11 R of A a11d i11jormatio11 abow cenai11 co/wm1s of A. 100874

/11 Exercises 67- 74, find the matrix A, given the retluced row echelon jon11 R of A a11d i11jormatio11 abow cenai11 co/wm1s of A. r = [~

/11 Exercises 67- 74, find the matrix A, given the retluced row echelon jon11 R of A a11d i11jormatio11 abow cenai11 co/wm1s of A. r = [~

/11 Exercises 67- 74, find the matrix A, given the retluced row echelon jon11 R of A a11d i11jormatio11 abow cenai11 co/wm1s of A. r = [~

/11 Exercises 67- 74, find the matrix A, given the retluced row echelon jon11 R of A a11d i11jormatio11 abow cenai11 co/wm1s of A. r = [~

In Exercises 75-78. write the indicated colwnn of A=[~ -2 -1 -4 I I -6 0 6 as tl linear combination of the pnm co/um11r of A. 75. ~ 76. a 3 /11 E.1erciw,, 79-82, urite the indicmed wi1111111 of 8 ~ [ ; 0 I -3 - I :] - I 3 !! -I I -2 5 -6 0 - I 2 -3 as a lint'flr combination of tilt' pivot columns of 8. a2

In Exercises 75-78. write the indicated colwnn of A=[~ -2 -1 -4 I I -6 0 6 as tl linear combination of the pnm co/um11r of A. 75. ~ 76. a 3 /11 E.1erciw,, 79-82, urite the indicmed wi1111111 of 8 ~ [ ; 0 I -3 - I :] - I 3 !! -I I -2 5 -6 0 - I 2 -3 as a lint'flr combination of tilt' pivot columns of 8. a4

In Exercises 75-78. write the indicated colwnn of A=[~ -2 -1 -4 I I -6 0 6 as tl linear combination of the pnm co/um11r of A. 75. ~ 76. a 3 /11 E.1erciw,, 79-82, urite the indicmed wi1111111 of 8 ~ [ ; 0 I -3 - I :] - I 3 !! -I I -2 5 -6 0 - I 2 -3 as a lint'flr combination of tilt' pivot columns of 8. a5

In Exercises 75-78. write the indicated colwnn of A=[~ -2 -1 -4 I I -6 0 6 as tl linear combination of the pnm co/um11r of A. 75. ~ 76. a 3 /11 E.1erciw,, 79-82, urite the indicmed wi1111111 of 8 ~ [ ; 0 I -3 - I :] - I 3 !! -I I -2 5 -6 0 - I 2 -3 as a lint'flr combination of tilt' pivot columns of 8. a6

/11 E.1erciw,, 79-82, urite the indicmed wi1111111 of 8 ~ [ ; 0 I -3 - I :] - I 3 !! -I I -2 5 -6 0 - I 2 -3 as a lint'flr combination of tilt' pivot columns of 8. b3

/11 E.1erciw,, 79-82, urite the indicmed wi1111111 of 8 ~ [ ; 0 I -3 - I :] - I 3 !! -I I -2 5 -6 0 - I 2 -3 as a lint'flr combination of tilt' pivot columns of 8. b4

/11 E.1erciw,, 79-82, urite the indicmed wi1111111 of 8 ~ [ ; 0 I -3 - I :] - I 3 !! -I I -2 5 -6 0 - I 2 -3 as a lint'flr combination of tilt' pivot columns of 8.b5

/11 E.1erciw,, 79-82, urite the indicmed wi1111111 of 8 ~ [ ; 0 I -3 - I :] - I 3 !! -I I -2 5 -6 0 - I 2 -3 as a lint'flr combination of tilt' pivot columns of 8. b6

Suppose that u and v are linearly independent vectors in 'R3. Find the reduced row echelon form of A = la1 3 2 3J a.). given that a , = u. 3 2 = 2u , 33 = u + v. and a.= v.

Let A be an " x n invenible matrix, and let CJ be the jth standard vector of 'R". (a) Prove that the j th column of A 1 is a solution of Ax = e 1. (b) Why does the result in (a) imply that ;\ 1 is unique? (c) Why doe.- the rCMih in (:~) imply th:tt r:mk A = 11?

Let A be a matrix with reduced row echelon form R. Usc the column correspondence propeny to prove the following: (a) A column of A is 0 if and only if the corresponding column of R is 0 . (b) A set of columns of A is linearly independent if and only if the corresponding 'et of columns of R is line:~rl y independent.

Let R be an 111 x 11 matrix in reduced row echelon form. Find a relationship between the columns of Rr and the columns of Rr R. Justify your answer.

Let R be an 111 x 11 matrix in reduced row echelon form with rank R = r. Prove the following: (a) The reduced row echelon form of Rr is the 11 x m matrix [e, Cl ... e, 0 . or. \\here e, i' the jth standard vector of n for I ~ j ~ r. (b) rank RT =rank

Let A be an m x n matrix with reduced row echelon form R. Then there exists an invcnible matrix P \Uch that PA = R and an invertible matrix Q ;,uch that QRT is the reduced row echelon form of RT. De;,cribe the matrix PAQT in terms of A. Justify your answer

Let A and 8 be 111 x 11 matrices. Prove that the following ccmdition' arc cquiv:~ nt. (a) A tmd 8 have the same reduced row echelon form. (b) There is an invertible m x m matrix fJ such that 8 = ~1.

Let A be an 11 x 11 matrix. Find a property of A that i' equivalent to the 'tatemcnt A8 = AC if and only if 8 = C. Justify your answer.

Let A be a 2 x 3 matrix, and let E be an elementary matrix obtained by an elementary row operation on /2. Pro\e that the product A can be obtained from A by the tdenllcal elementary row operation on A. Him: Prove thi;, for

Let A and 8 be m x 11 matrices. Prove that the following conditions are equivalent: (a) A and 8 have the same reduced row echelon form. (b) The system of equations Ax = 0 is equivalent to 8x = 0.

Let f:.' be an 11 x 11 matrix. Prove that E is an elementary matrix if and o nly if can be obtained from /11 by a single e lement ary column operution.

Prove that if tt matrix E i> oblinglc e lementary column operation, then for any m x 11 matrix A. the product A can be obtained from A by the identical elementary column operation on A.

In E.urift'f 95 99. liSt' eithu a calculator with lllfltrit rapa bilities or compmer sofna~ such as MATU\8 to solte elicit problmr. Let Let 8 be the matrix obtained

In E.urift'f 95 99. liSt' eithu a calculator with lllfltrit rapa bilities or compmer sofna~ such as MATU\8 to solte elicit problmr. Let [ 15 A= 30 17 31 30 17 66 36 36 20 6 1 35 61 3 1] 35 . 65 (a) Show that A is ymmetric and invertible. (b) Show that A- t is ymmetric and invertible. (c) Prove that the inverse of any s

In E.urift'f 95 99. liSt' eithu a calculator with lllfltrit rapa bilities or compmer sofna~ such as MATU\8 to solte elicit problmr. Let 2 5 4 6 0 -I ll (a) Show that A and A 2 are invertible by using their reduced row eche lon forms. (b) Compute lhe inverse of A2, and show that it equals (A- 1)2.

In E.urift'f 95 99. liSt' eithu a calculator with lllfltrit rapa bilities or compmer sofna~ such as MATU\8 to solte elicit problmr. Consider the system Ax = b. Xt + 3X2 +

In E.urift'f 95 99. liSt' eithu a calculator with lllfltrit rapa bilities or compmer sofna~ such as MATU\8 to solte elicit problmr. The purpose of this exercise is to illustrate the method of finding an inverse described in Exercise 84. In Exercise 97, you found the inverse of