Solved: In rercises 19- 26, use rl1e algtrithm for

In Eurcises 1- 18, detennine whether each matrix is inrertible.and If so. find its imerse 132

In Eurcises 1- 18, detennine whether each matrix is inrertible.and If so. find its imerse 1242

In Eurcises 1- 18, detennine whether each matrix is inrertible.and If so. find its imerse 1326

In Eurcises 1- 18, detennine whether each matrix is inrertible.and If so. find its imerse _;

In Eurcises 1- 18, detennine whether each matrix is inrertible.and If so. find its imerse ;]

In Eurcises 1- 18, detennine whether each matrix is inrertible.and If so. find its imerse -

In Eurcises 1- 18, detennine whether each matrix is inrertible.and If so. find its imerse [:

In Eurcises 1- 18, detennine whether each matrix is inrertible.and If so. find its imerse [:

In Eurcises 1- 18, detennine whether each matrix is inrertible.and If so. find its imerse [:

In Eurcises 1- 18, detennine whether each matrix is inrertible.and If so. find its imerse [:

In Eurcises 1- 18, detennine whether each matrix is inrertible.and If so. find its imerse [:

In Eurcises 1- 18, detennine whether each matrix is inrertible.and If so. find its imerse [:

In Eurcises 1- 18, detennine whether each matrix is inrertible.and If so. find its imerse [:

In Eurcises 1- 18, detennine whether each matrix is inrertible.and If so. find its imerse [:

In Eurcises 1- 18, detennine whether each matrix is inrertible.and If so. find its imerse [:

In Eurcises 1- 18, detennine whether each matrix is inrertible.and If so. find its imerse [:

In Eurcises 1- 18, detennine whether each matrix is inrertible.and If so. find its imerse [:

In Eurcises 1- 18, detennine whether each matrix is inrertible.and If so. find its imerse [:

In rercises 19- 26, use rl1e algtrithm for comiHifing A_ , 8 9. [~ i] 8 -

In rercises 19- 26, use rl1e algtrithm for comiHifing A_ , 8 20. A = [-1

In rercises 19- 26, use rl1e algtrithm for comiHifing A_ , 8 A= [; ~] a

In rercises 19- 26, use rl1e algtrithm for comiHifing A_ , 8 ~ -1 n 8 = [! -2] 3. 4. - 1

In rercises 19- 26, use rl1e algtrithm for comiHifing A_ , 8 2 3 ~] 8 = [! 0 1 23. A= -: -n 1

In rercises 19- 26, use rl1e algtrithm for comiHifing A_ , 8 A = [ : 2 ~] and 8 = [i -I 0 2 - 1 3 -I 2 4 -~] 2

In rercises 19- 26, use rl1e algtrithm for comiHifing A_ , 8 and I 2~. A = ~ 0 I -I - I 0 [:- 1 j] [l - I ] 0 0

In rercises 19- 26, use rl1e algtrithm for comiHifing A_ , 8 and I 2~. A = ~ 0 I -I - I 0 [: - 1 j] [l - I ] 0 0

In Erercises 27- 34, a matrix A is given. Determine (a) the reduced row echelon form R of A and (b) an invertible matrix P such thai ~\ = R. ~ -I

In Erercises 27- 34, a matrix A is given. Determine (a) the reduced row echelon form R of A and (b) an invertible matrix P such thai ~\ = R. 11021

In Erercises 27- 34, a matrix A is given. Determine (a) the reduced row echelon form R of A and (b) an invertible matrix P such thai ~\ = R. {{{}}}

In Erercises 27- 34, a matrix A is given. Determine (a) the reduced row echelon form R of A and (b) an invertible matrix P such thai ~\ = R. {1238}

In Erercises 27- 34, a matrix A is given. Determine (a) the reduced row echelon form R of A and (b) an invertible matrix P such thai ~\ = R. {1238}

In Erercises 27- 34, a matrix A is given. Determine (a) the reduced row echelon form R of A and (b) an invertible matrix P such thai ~\ = R. {1238}

In Erercises 27- 34, a matrix A is given. Determine (a) the reduced row echelon form R of A and (b) an invertible matrix P such thai ~\ = R. {1238}

In Erercises 27- 34, a matrix A is given. Determine (a) the reduced row echelon form R of A and (b) an invertible matrix P such thai ~\ = R. {1238}

A matrix is invertible if and o nly if its reduced row echelon form is an identity matrix

For any two matrices A and B, if A8 = 1, for some positive integer 11 , then A is invertible

For any two 11 x n matrices A and B. if AB = 1,, then BA = 1

For any two 11 x 11 matrices A and B, if AB = 1,, then A is invettible and A- t = B

If an 11 x 11 matrix has rank 11 . then it is invertible.

If an 11 x n matrix is invertible. then it has rank 11.

A square matrix is invertible if and only if its reduced row echelon fonn has no zero row

lf A is an 11 x 11 mattix such that the onl y solution of Ax = 0. then A is in vertible.

An 11 x 11 matrix is invettible if and o nly if its columns are linearly independent

An 11 x 11 matrix is invertible if :md only if its rows arc linearly independent

If a square matrix has a column consisting of all zeros. then it is not invertible

If a square matrix has ;.-~ row consisting of all zeros. 1hcn it is not invertible.

Any invertible matrix can be wrillen as a product o f e lementary matrices.

f A and 8 a re invertible 11 x n matrices. then A + 8 is invertible.

If A is :m n x " matrix s uch that Ax = b is consistent for ever)' b in 'R." , then Ax = b has a unique solution for every b in n

If A is an invertible 11 x 11 matrix and the reduced row eche lon fom1 of lA B ] is [/, C ], then C = s - tA.

If the reduced row echelon fonn of [A /,] is [R B]. then 8 =A- 1

e reduced row echelon form of [A /, ]is [R 8]. then B is an invertible matrix.

If the reduced row echelon form of [A 1, ] is [R 8], then BA equals the reduced row echelon form of

Suppose that A is an invertible matrix and u is a solution of A< = m TI" ol,foo of A< = m dfff= rn>o by 2p 3, where P3 is the third column o f A- 1

Prove directly that statement (a) in the Invertible Matrix Theorem implies statements (e) and (h).

In Exercises 56- 63, a sysrem of linear equarions is given. (a) Wrire each sysrem as ammrix equarion Ax = b. (b) Show rlwr A is invertible. and find A- 1 (c) Use A- 1 ro solve each sysrem. t + 2x2 =

In Exercises 56- 63, a sysrem of linear equarions is given. (a) Wrire each sysrem as ammrix equarion Ax = b. (b) Show rlwr A is invertible. and find A- 1 (c) Use A- 1 ro solve each sysrem. -x1 - 3x2 = -6

In Exercises 56- 63, a sysrem of linear equarions is given. (a) Wrire each sysrem as ammrix equarion Ax = b. (b) Show rlwr A is invertible. and find A- 1 (c) Use A- 1 ro solve each sysrem. + 2xz + 6x

In Exercises 56- 63, a sysrem of linear equarions is given. (a) Wrire each sysrem as ammrix equarion Ax = b. (b) Show rlwr A is invertible. and find A- 1 (c) Use A- 1 ro solve each sysrem. 3 = 3

In Exercises 56- 63, a sysrem of linear equarions is given. (a) Wrire each sysrem as ammrix equarion Ax = b. (b) Show rlwr A is invertible. and find A- 1 (c) Use A- 1 ro solve each sysrem. +x3 =

In Exercises 56- 63, a sysrem of linear equarions is given. (a) Wrire each sysrem as ammrix equarion Ax = b. (b) Show rlwr A is invertible. and find A- 1 (c) Use A- 1 ro solve each sysrem. +x3 =

In Exercises 56- 63, a sysrem of linear equarions is given. (a) Wrire each sysrem as ammrix equarion Ax = b. (b) Show rlwr A is invertible. and find A- 1 (c) Use A- 1 ro solve each sysrem. +x3 =

In Exercises 56- 63, a sysrem of linear equarions is given. (a) Wrire each sysrem as ammrix equarion Ax = b. (b) Show rlwr A is invertible. and find A- 1 (c) Use A- 1 ro solve each sysrem. +x3 =

Let A = [ : a (a) Verify that A2 - 3A + l2 = 0. (b) Let B = 3/2 -A. Use 8 to prove that A is in vertible and 8 = A- 1

t A = [ _ ~ - ~ _ n (a) Verify that A3 - 5A2 + 9A- 413 = 0. I (b) Let 8 = ;j(A2 - 5A + 9/j). Use 8 to prove that A is invertible and 8 =A - I. (c) Explain how 8 in (b) can be obtained from the equation in (a)

Let A be an 11 x 11 matrix such that A 2 "" 1, . Prove that A is invertible and A- 1 =A.

Let A be an n x 11 matrix such that AA = 1. for some posttl\e integer k. (a) Prove that A is invertible.

Prove that if A is an m x 11 matrix and P is an invertible 111 x m matrix. then rank P

Let 8 be an 11 x p matrix. Prove the fol lowing: (a) For any m x 11 matrix R in reduced row echelon fonn. rank R8 !:: rank R. Him: Usc the definition of the reduced row echelon fonn. (b) If A is any m x 11 matrix. then rank A8 !:: rank A. /lim: Use Exercises 68 and 69(a).

Prove that if A is an m x 11 matrix and Q is an imertible 11 x 11 matrix. then rank AQ = rank

Prove that for any matrix A. rank AT= rank A. Him: u,e Exercise 70. Exerci'e 87 of Section 2.3, and Theorems 2.2 and 2.3

Usc the In vertible Matrix Theorem to prove that for any subsetS of 11 vectors in R.". the setS h. linearly independent if and only if S is a generating set for nn.

Let R and S be matrices with the same number of rows. and suppose that the matrix [R S I is in reduced row echelon fonn. Pro,e that R is in reduced row echelon fonn.

Consider the system of linear equations Ax = b. where A= [i 2 3 4 and [ 20] b = !~ . (a) Solve this matrix equation by using Gaussian eliminatlon. (b) On a TJ-85 calculator. the value of A_, b is given as But this is 11ot a solution of Ax =

Repeat Exercise 74 with 2 3 7

Repc:tt Exercise 74 with A = [i 2 5 8

In Exercise 19(c) of Section 1.5. ho" much is required in additional inputs from each sector to increa.~e the net production of oil by $3 million?

In Exerci\e 20(c) of Section 1.5. how much is required in addition:tl inputs from each sector to increase net production in the nongovernment sector by S I million?

In Exercise 21(b) of Section 1.5. how much is required in additional inputs from each sector to increase the net production of scrvtces by $40 million?

In Exercise 22(b) of Section 1.5. how much is required in udditional inputs from each sector to increase the net production of manufacturing by $24 million?

Suppo;.e that the input- output matrix C for an economy i> ;.uch that 1. - C is invenible and every entry of (1. -C) t is positive. If the net production of one panicular sector of the economy must be increased. how does this affect the gross production of the economy?

U;c matnx transposes to modify the algonthm for computing A - 18 to devise an algorithm for computing A8 - t. and JU'tify your method.

Let A be un m x n matrix with reduced row echelon form R. (u) Prove that if rank A = m, then there is u unique m x m matrix P such that PA = R. Furthermore. I' is invertible. /lim: For each j. let u; denote the jth pivot column of A. Prove that the m x m matrix U = (u 1 u 1 . . Um] is invertible. Now let PA = R. and show that P = u-t. (b) Pro' e that tf rank A < m. Lben there is more than one invcnible m x m matrix P such that PA = R. flint: There is an elementary m x m matrix E. distinct from 1,.. ;.uch that ER = R.

Let A, fl. :tnd C be 11 x 11 matrices. Pwvc the following statements: (a) II is similar to A. (b) If II is similar to 8, then 8 is similar to A. (c) If II is similar to 8 and 8 is similar to C. then A is Mmilar to C

Let A be an II X II matrix. (a) Prove that if A is similar to 1 . then A = 1 . (b) Pro'e that if A is similar to 0. the 11 x 11 t.ero matrix. then A= 0. (c) Suppose thut 8 = cl., for some calar 1. (l11e mutrix 8 is called u ~calor matri..r.) Whut can you suy about A if A is simihtr to 8 ?

Suppose that A and 8 are 11 x 11 mutrices such that A is similar to 8 . Prove that if A is inveniblc. then 8 is invertible. and 11 - t is similar to 8 - 1.

Suppo-c that A and 8 arc 11 x n matrice' such th:ll ;\ h similar to 8 . Prove that AT is similar to 8 T.

Suppo-c that A and 8 are n x 11 matrices 'uch that A is ~imtlar to B. Pro\e that rank A = rank B. /lim: Use Exerci>es 68 and 70.

In Exercises 89- 92. use either a calcularor with marrix capabiliries or compwer softwarr! such os MATI.A8 ro solve each problem. Exercises 89- 9 I refer to rhe matrix

In Exercises 89- 92. use either a calcularor with marrix capabiliries or compwer softwarr! such os MATI.A8 ro solve each problem. Exercises 89- 9 I refer to rhe atrix 5 8 6 9 6 9 5 7 l Show that A is invertible y c-omputing its reduced row echelon form and using Theorem 2.5

In Exercises 89- 92. use either a calcularor with marrix capabiliries or compwer softwarr! such os MATI.A8 ro solve each problem. Exercises 89- 9 I refer to rhe matrix 5 8 6 9 6 9 5 7 l Show that A is invertible by computing its rank and using

In Exercises 89- 92. use either a calcularor with marrix capabiliries or compwer softwarr! such os MATI.A8 ro solve each problem. Exercises 89- 9 I refer to hematrix 5 8 6 9 6 9 5 7 l Show that the matrix 2 3 4 6 -1 2 -1 -