Let u be a solution of Ax = 0. where A i' an m x n matrix.

In ..\ercises I /6, dt'll'nniltt' whether the gil'en system ;,,.con .sisrem. and if so, find irs ge11eml so/111io11. 2x1 + Ci.rz = - 4

In ..\ercises I /6, dt'll'nniltt' whether the gil'en system ;,,.con .sisrem. and if so, find irs ge11eml so/111io11. l + 2x2 = -6

In ..\ercises I /6, dt'll'nniltt' whether the gil'en system ;,,.con .sisrem. and if so, find irs ge11eml so/111io11. l.r1 + 3

In ..\ercises I /6, dt'll'nniltt' whether the gil'en system ;,,.con .sisrem. and if so, find irs ge11eml so/111io11. 2.r1 + x2 -

In ..\ercises I /6, dt'll'nniltt' whether the gil'en system ;,,.con .sisrem. and if so, find irs ge11eml so/111io11. 11 - 2x2 + 4xl = - 411 + 412- 81) = -3

In ..\ercises I /6, dt'll'nniltt' whether the gil'en system ;,,.con .sisrem. and if so, find irs ge11eml so/111io11. -'1 - 21! - XJ- 3

In ..\ercises I /6, dt'll'nniltt' whether the gil'en system ;,,.con .sisrem. and if so, find irs ge11eml so/111io11. -'1 - 2.tl - -'l = -3

In ..\ercises I /6, dt'll'nniltt' whether the gil'en system ;,,.con .sisrem. and if so, find irs ge11eml so/111io11. x1 + xz "' x. = 2

In ..\ercises I /6, dt'll'nniltt' whether the gil'en system ;,,.con .sisrem. and if so, find irs ge11eml so/111io11. 9. -l.r1 + X2 + 5.n = -4

In ..\ercises I /6, dt'll'nniltt' whether the gil'en system ;,,.con .sisrem. and if so, find irs ge11eml so/111io11. \I - J.\2 + XJ + X4 = 0 10. -3xl + 9xl- 2rJ- 5.q = 2q - 6x2 - .n + sx. = -2

In ..\ercises I /6, dt'll'nniltt' whether the gil'en system ;,,.con .sisrem. and if so, find irs ge11eml so/111io11. \ 1 + 3..12 + fJ + t4 = - I

In ..\ercises I /6, dt'll'nniltt' whether the gil'en system ;,,.con .sisrem. and if so, find irs ge11eml so/111io11. 211+ .\2 + .\.\=-I\!- X.\= 2 - lr3

In ..\ercises I /6, dt'll'nniltt' whether the gil'en system ;,,.con .sisrem. and if so, find irs ge11eml so/111io11. X1 + 2x2 + X! = I13 _ -2.r1 - 4xz - XJ = 0

In ..\ercises I /6, dt'll'nniltt' whether the gil'en system ;,,.con .sisrem. and if so, find irs ge11eml so/111io11. x1 - x2 +x.l = 7 14 _ x1 - 2.r2 - r3 = 8

In ..\ercises I /6, dt'll'nniltt' whether the gil'en system ;,,.con .sisrem. and if so, find irs ge11eml so/111io11. 1 - ' J - lr4 - &.15 = -3

In ..\ercises I /6, dt'll'nniltt' whether the gil'en system ;,,.con .sisrem. and if so, find irs ge11eml so/111io11. .r1 - 12 + \4 = -4

/11 E.urcises 17 26. detem1i11e the trlues of r. if lliiJ. for ..-hich the gie11 system of lillt'ilf equati1111.r i.r iiiWIUiltell/. 7. -x1

/11 E.urcises 17 26. detem1i11e the trlues of r. if lliiJ. for ..-hich the gie11 system of lillt'ilf equati1111.r i.r iiiWIUiltell/. 18. 3x1 + r.t2 = - 23x1 + rxz = 2 -r1 + 4.12

/11 E.urcises 17 26. detem1i11e the trlues of r. if lliiJ. for ..-hich the gie11 system of lillt'ilf equati1111.r i.r iiiWIUiltell/. x1

/11 E.urcises 17 26. detem1i11e the trlues of r. if lliiJ. for ..-hich the gie11 system of lillt'ilf equati1111.r i.r iiiWIUiltell/. XI+ f.\2

/11 E.urcises 17 26. detem1i11e the trlues of r. if lliiJ. for ..-hich the gie11 system of lillt'ilf equati1111.r i.r iiiWIUiltell/. XI - 3x2 = -2

/11 E.urcises 17 26. detem1i11e the trlues of r. if lliiJ. for ..-hich the gie11 system of lillt'ilf equati1111.r i.r iiiWIUiltell/. -2.rl + .Q =5

/11 E.urcises 17 26. detem1i11e the trlues of r. if lliiJ. for ..-hich the gie11 system of lillt'ilf equati1111.r i.r iiiWIUiltell/. 23. -x1

/11 E.urcises 17 26. detem1i11e the trlues of r. if lliiJ. for ..-hich the gie11 system of lillt'ilf equati1111.r i.r iiiWIUiltell/. 24. .II

/11 E.urcises 17 26. detem1i11e the trlues of r. if lliiJ. for ..-hich the gie11 system of lillt'ilf equati1111.r i.r iiiWIUiltell/. 25. Xt - .r2 + 2r3 = 4 26. X1 + 2.r2 - 4XJ = I

/11 E.urcises 17 26. detem1i11e the trlues of r. if lliiJ. for ..-hich the gie11 system of lillt'ilf equati1111.r i.r iiiWIUiltell/. 3

/11 E.rercises 27-34, de/ermine the talues ofr cmd sfarwhich the gien sysrem af li11ear eqtwtimt< lw< (II) no .rolurio11s. (b) exactly one solwion. 11nd (c) infinirely "'""-"so/Ill ions. 27 .r1

/11 E.rercises 27-34, de/ermine the talues ofr cmd sfarwhich the gien sysrem af li11ear eqtwtimt< lw< (II) no .rolurio11s. (b) exactly one solwion. 11nd (c) infinirely "'""-"so/Ill ions. -.II

/11 E.rercises 27-34, de/ermine the talues ofr cmd sfarwhich the gien sysrem af li11ear eqtwtimt< lw< (II) no .rolurio11s. (b) exactly one solwion. 11nd (c) infinirely "'""-"so/Ill ions. 29. X

/11 E.rercises 27-34, de/ermine the talues ofr cmd sfarwhich the gien sysrem af li11ear eqtwtimt< lw< (II) no .rolurio11s. (b) exactly one solwion. 11nd (c) infinirely "'""-"so/Ill ions. = s

/11 E.rercises 27-34, de/ermine the talues ofr cmd sfarwhich the gien sysrem af li11ear eqtwtimt< lw< (II) no .rolurio11s. (b) exactly one solwion. 11nd (c) infinirely "'""-"so/Ill ions. 2xt

/11 E.rercises 27-34, de/ermine the talues ofr cmd sfarwhich the gien sysrem af li11ear eqtwtimt< lw< (II) no .rolurio11s. (b) exactly one solwion. 11nd (c) infinirely "'""-"so/Ill ions. Xt + f

/11 E.rercises 27-34, de/ermine the talues ofr cmd sfarwhich the gien sysrem af li11ear eqtwtimt< lw< (II) no .rolurio11s. (b) exactly one solwion. 11nd (c) infinirely "'""-"so/Ill ions. x1 +

/11 E.rercises 27-34, de/ermine the talues ofr cmd sfarwhich the gien sysrem af li11ear eqtwtimt< lw< (II) no .rolurio11s. (b) exactly one solwion. 11nd (c) infinirely "'""-"so/Ill ions. x1 + r

In Exercise.< 35-42.jind 1/re rank tmd 1wlli1y of 1he given mtllrix. 35. [-j

In Exercise.< 35-42.jind 1/re rank tmd 1wlli1y of 1he given mtllrix. j -3 - I -l] 6 2

In Exercise.< 35-42.jind 1/re rank tmd 1wlli1y of 1he given mtllrix. ~ 2 I -I -I -3- I - I

In Exercise.< 35-42.jind 1/re rank tmd 1wlli1y of 1he given mtllrix. 38. {{

In Exercise.< 35-42.jind 1/re rank tmd 1wlli1y of 1he given mtllrix. [i I ] 2 4 0 - 4 39

In Exercise.< 35-42.jind 1/re rank tmd 1wlli1y of 1he given mtllrix. -l -I 5 -I

In Exercise.< 35-42.jind 1/re rank tmd 1wlli1y of 1he given mtllrix. [ -l -2 0 -3 - 4 - I - 8 2 5 0 2 4

In Exercise.< 35-42.jind 1/re rank tmd 1wlli1y of 1he given mtllrix. l -2 - I 0 -4 -2 -I 2 I 0 0 4

A mining company operates three mines that each produce three grades of ore. The daily yield of e"ch mine is shown in the following table: Daily Yield Mme I Mme2 Mine J llighgradc ore I ton I ton 2 to ns \1edium-gratlc ore 1 ton 2 to ns 2 to ns Low-grade ore 2 tons I ton 0 tons (a) Can the company supply exactly 80 tons of highgrade. 100 tons of medium-grade. and 40 tons of

A company makes three types of fertil izer. The first type contains 10% nitrogen and 3% phosphates by weight, the second contains 8% nitrogen and 6% phosphates, and the third contains 6% nitrogen and I% phosphates. (a) Can the company mix these three types of fertilizers to supply exactly 600 pounds of fertil izer containing 7.5% nitrogen and 5% phosphates? If so, how? (b) Can the company mix these three types of fertilizers to supply exactly 600 pounds of fertil izer containing 9% nitrogen and 3.5% phosphates? If so, how?

A patient needs to consume exactly 660 mg of magnesium. 820 IU of vitamin D. and 750 meg of folate per day. Three food supplements can be mixed to provide these murients. The amounts of the three nutrients provided by each of the supplements is given in the following table: 46. Food Supplement I 2 3 Magne>ium (mg) 10 15 36 Vitamin D (IU) 10 20 44 Fo late (meg) 15 15 42 (a) What is the maximum amount o f supplement 3 that can be used to provide exactly the required amounts of the three nutrients? (b) Can the three supplements be mixed to provide exactly 720 mg of magnesium. 800 IU of vitamin D. and 750 meg of folate? If so. how?

Three grades of crude o il :ue to be blended to obtai n 100 barrels of oil costing $35 per barrel and containing 50 gm of sulfur per barrel. The cost and sulfur content of the three grades of oil are given in the following table: Co>t per barrel Sulfur per barrel Grade B $32 62 gm c $24 94 gm (a) Find the amounts of each grade to be ble nded that use the least oil of grade C. (b) Find the amounts of each grade to be blended that use the most oil of grade C

Find a polynomial function f(x) = ax 2 + bx + c whose graph passes through the points (- I. 14). ( 1.4). and (3. 10)

Find a polynomial function f(x) = a.~ 2 + bx + c whose gmph passes through the points ( - 2, -33). (2. -I), :md (3,-8)

Find a polynomial function f(x) = ttxl + bx2 +ex+ d whose graph passes through the points (-2, 32). (-1, 13), (2.4), and (3. 17).

Find a polynomial function f(x) = ar3 + bx2 +ex+ d whose gruph passes through the points (-2. 12). (- 1. - 9). (I. -3). and (3. 27

If the thrrd pivot posruon of a matrix A is m column j. what can be sard about column j of the reduced row echelon fonn of A? Explain your answer

Suppo-.c that the founh pivot po

A column of a matrix A is a pivot column if the corresponding column in the reduced row echelon form of A contains the leading entry of some nonzero row.

There is a unique sequence of elementary row operations that transforms a matrix into its reduced row echelon form

When the forward pass of Gaussian elimination is complete. the origimrl matrix has been transformed into one in mw echelon fonn.

No ;,caling opcrutions arc re

The rank of a matrix equal< the number of pivot columns in the matrix.

If Ax = b is conshtent. then the nullity of A equals the number of free variable; in the gcner.tl solution of Ax = b.

There cxi>t' a 5 x 8 matrix with runk 3 and nullity 2.

If a ;ystem of m linear equations in 11 variables is equivalent to a ' )'>tem of 11 linear equation.' in q variable>. then "'=fl.

If a S)"Stem of m linear equations in II variables is equi\'3 lent to a system or p linear equations in q variables. then II =q.

The equation Ax = b s consistent if and only if b is a linear combrnatron of the column; of A.

If the equation Ax = b ;, inconsi~tent. then the rank of [A bJ is greater than the rank of A.

If the reduced row echelon form of lA b) contains a zero row. then Ax = b mu>t have infinitely many solutions.

If the reduced row echelon form of lA bl contains a zero row. then Ax = h mu>t be con>istent.

If some column of m:1trix A is a pivot column. then the corresponding column in the reduced row echelon form of A is a ;tandurd vector.

If A is a matrix with r:u1k k. then the vectors e1 e2 et :rppe;rr as columns of the reduced row echelon form of A.

The surn of the rank and nullity of a matrix equals the number of rows in the matrix.

Suppose that the pivot ows of a matrix A :~re rows I. 2 ..... k. and row k + I becomes zero when applying the Gaussian elimination algorithm. Then row k + I must equal borne linear combination of rows I. 2 ..... k,

The third pivot po;ition in a matrix lies in row 3.

The third pivot position in a matrix lies in column 3.

If R is an 11 x 11 matrix in reduced row echelon form that has rank" then R = 1

Describe an m x 11 matrix with rank 0.

What is the smallest possible rank of a 4 x 7 matrix? Explain your answer

What is the largest possible rank of a 4 x 7 matrix? Explain your answer.

What is the la.rgest possible rank of a 7 x 4 matrix? Explain your answer.

What is the smallest possrble nullrty of a 4 x 7 matrix? Explain your answer

What is the smallest possible nullity of a 7 x 4 matrix? Explain your an>wer.

What is the largest possible rank of an m x 11 matrix? Explain your answer

What is the smallest possible nullity of an m x 11 matrix? Explain your answer

Let A be a 4 x 3 matrix. Is it possible that Ax = b is consistent for every b in n? Explain your an>wcr.

Let A be an m x 11 matrix and b be a vector in n"'. What must be true ahout the rank of A if Ax = b has a unique solution? Justify your ;mswcr.

A >ystern of linear equations is called wrdertletermi11ed if it has fewer equation; than variables. What can be said ahout the number of solutions of an underdetennined system'!

A system of linear equations is called Ol'etrletumill<'d if it has more equations than variables. Gi,e examples of overdetermined 'Y'tem' that ha\e (a) no >Oiution>. (b) exactly one solution. and (c) infinitely many solullons.

Prove that if A is an m x 11 matrix with rank m. then AX = b b con'i'tent for ecry b in n,..

Prove that a matrix equation Ax = b is consi>tent if and only if the rank of A and [A bJ arc equal.

Let u be a solution of Ax = 0. where A i' an m x n matrix. Must c u be a solution of Ax = 0 for every scalar c? Justify your answer.

Let u and v be solutions of Ax = 0. where A is an m x 11 matrix. Must u + v be a solution of llx = 0? Justify your answer.

Let u and ' ' be solution$ of Ax = b. where A is ~u1 m x 11 matrix and b is a vector in nm. Prove that u - v is a solutio n of Ax = 0.

Let u be a solution of Ax = b and v be a solution of Ax = 0. where A is an m x 11 matrix :md b i' a vector in n'". Prove that u + v is a solution of Ax = b.

Let II be an m x 11 matrix and b be a vector in n'" such that Ax = b i~ con~ist enl. Prove that Ax = cb is consistent for e\ery scalar c

Let A be an m x 11 matrix and b 1 and b2 be vectors in n"' such that both Ax = h t and Ax = b2 are consistent. Prove that Ax = bt + h2 is consistent.

Let u and v be solutions of Ax = b, where A is

/11 Exercises 94- 96. use Gaussia11 e/imi11atio11 011 the augmemed mmrix of the sysrem of linear equmio11s to test for consisTency, cmd to ji11d the general solution. 94. 2.2xt - 4

/11 Exercises 94- 96. use Gaussia11 e/imi11atio11 011 the augmemed mmrix of the sysrem of linear equmio11s to test for consisTency, cmd to ji11d the general solution. 95_ 2Xt + X

/11 Exercises 94- 96. use Gaussia11 e/imi11atio11 011 the augmemed mmrix of the sysrem of linear equmio11s to test for consisTency, cmd to ji11d the general solution. 4Xt- X2 + 5XJ

/11 Exercises 97- 99, ji11d the rank and the nul/if) of the mmrix. ,, 2.3 -1.1 1.0 ''

/11 Exercises 97- 99, ji11d the rank and the nul/if) of the mmrix. ' 1 1.3 1.6 1.5 1.7 2.3 -1.2 2.1 -LO] 2.2 98. 3. 1 -1.

/11 Exercises 97- 99, ji11d the rank and the nul/if) of the mmrix. - II 2 4 -:] I 0 8