Questions on Algebra: Matrices, determinant, Cramer rule answered by real tutors!

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Question 615967: We're trying to help our soon with his math homework but we're totally stuck one this problem:
A store has a sale on almonds, pecans and pistachios. One lb of almonds, one lb of pecans and one lb of pistachios cost $12. Two lbs of almonds and three lbs of pecans cost $16. Three lbs of pecans and two lbs pistachios cost $24. Find the price of each kind of nut.

Thank you in advance for any help.
Click here to see answer by ikleyn(53213)

Question 1159507: Find a non-zero 2×2 matrix such that:
[-9, -3] * [] = [0, 0]
[18, 6] [] [0, 0]
Click here to see answer by ikleyn(53213)

Question 1159895: Find a basis of the subspace of R^4 defined by the equation −5x1−5x2−4x3−8x4=0.
Click here to see answer by ikleyn(53213)

Question 1159842: Find a basis of the subspace of R4 defined by the equation −5x1−5x2−4x3−8x4=0.
Click here to see answer by ikleyn(53213)

Question 1165621: Solve the system:
{4x1+x2=7
{12x1+3x2=21
[x1]=[]+s[]
[x2]=[]+s[]
Click here to see answer by ikleyn(53213)

Question 1168173: Lanre income from interest and wages is 4000 he double his investment and also get an increase to 6500 my what was his original income separately in both term (interest and wages). Using crammer rules.

Click here to see answer by CPhill(2103)

Question 1169744: The general linear supply and demand for one-commodity market model are given by
P = a𝑄𝑠+b (a>0, b>0)
P = - c𝑄𝐷+d (c>0, d>0)
a. Show that in matrix notation the equilibrium price, P, and quantity, Q satisfy.
[1 −𝑎] [𝑃]=[𝑏]
1 𝑐 𝑄 𝑑
b. Solve this system to express P and Q in terms of a, b, c and d.
c. Write down the multiplier for Q due to changes in b and deduce that an increase in b lends to an decrease in Q.

Click here to see answer by CPhill(2103)

Question 1171379: An architect has a pool in the floor plans for a home. Set up a determinant that gives the unit area of the pool. Coordinates given: (4,4) (1, -5) (-5, -3)
Click here to see answer by ikleyn(53213)

Question 1175738: If point A = matrices 4 -4 -3 and B = 2 0 2 . Then find AB and BA
0 0 -2 2 -2 1
1 -4 -2 -1 -1 -2
I tried the following. I am not sure how to get the other numbers.
4x2 + -4x2 + -3x-1=3
0x0 + 0x-2 + -2x-1=-2
1x2 + -4x1 + -2x-2=2

Click here to see answer by CPhill(2103)

Question 1177662: The verbiage in this question is confusing me so much that I don't even know where to start. Any help in explaining and solving would be greatly appreciated.
The following table shows the amount spent by four U.S. airlines to fly one available seat 1 mile in the second quarter of 2014. Set up a system and then solve using technology.
// The table shows:
Airline ==> United Continental - American - Jet Blue - Southwest
Cost(in cents)==> 14.9, 14.6, 11.9, 12.4

Suppose that, on a 3,000-mile New York to Los Angeles flight, United Continental, American, and Southwest flew a total of 235 empty seats, costing them a total of $99,060. If United Continental had three times as many empty seats as American, how many empty seats did each of these three airlines carry on its flight?
Click here to see answer by Edwin McCravy(20077)

Question 1178478: This question requires a confusion matrix. I am unsure how to calculate the accuracy given the percentages. How do I go about this? Thanks. :)
Both model A and model B achieves 80% accuracy on the balanced population, but changes to 64% and 96% accuracy respectively on the true unbalanced population. If the population is more skewed with 5% positive and 95% negative, what are the accuracies for each model?(The training population is 1,000 and the confusion matrix for model A is [500 200]
[0 300]
The confusion matrix for model B is [300 0]
[200 500]).

Click here to see answer by CPhill(2103)

Question 1181408: Given the following matrices:
b1 b4 b7
b2 b5 b8 = B
b3 b6 b9
r(B)= 3
For easier understanding let’s say that bi is the column i in matrix B.
A is a matrix of dimensions 3x3 in which the following exists:
K = { (b1+b3) , (2b2-b3) } is a basis for the equation: Ax=0.
r(A)=1
Also:
1 2 4
3 1 2 =A*B
1 2 4
r(AB)=2.

1) find a vector basis for the vector space that is A’s columns.
2) given the following:
K = { (1,2,0) , (0,1,1) }
Find a reduced echelon form (C) for A.

Click here to see answer by CPhill(2103)

Question 1181409: Given the following matrices over Z5:
b1 b4 b7
b2 b5 b8 = B
b3 b6 b9
r(B)= 3
For easier understanding let’s say that bi is the column i in matrix B.
A is a matrix of dimensions 3x3 in which the following exists:
K = { (b1+b3) , (2b2-b3) } is a basis for the equation: Ax=0.
r(A)=1
Also:
1 2 4
3 1 2 =A*B
1 2 4
r(AB)=2.

1) find a vector basis for the vector space that is A’s columns.
2) given the following:
K = { (1,2,0) , (0,1,1) }
Find a reduced echelon form (C) for A.
Click here to see answer by CPhill(2103)

Question 1182632: Given the representation matrix of a linear transformation over the basis B, how to find a basis for a different representation matrix for the same linear transformation in which a certain condition is required?
For example:
V = R2[x] (all the polynomials of power 2 or less over Real numbers)
Given the following basis:
B= {1, 1+x, x^2 -1}
Defining T: V -> V a linear transformation as following:
T(1) = 4 - 3x^2
T(1+x) = 4 + 4x + 3x^2
T(x^2 -1) = x^2 - 4
The representative matrix of T is:
1 3 -3
0 4 0 = [T]B
-3 3 1
If given that T is diagonalizable, and the eigenvalues of it are { 4, 4, -2}
Find a basis of V for which the representation matrix is a diagonal matrix.
Click here to see answer by CPhill(2103)

Question 1189235: DENTAL FILLINGS Dentists use various amalgams for silver fillings. The matrix shows the percent's (expressed as decimals) of powdered alloys used in preparing three different amalgams. Suppose a dentist has 5483 grams of silver, 2009 grams of tin, and 129 grams of copper. How much of each amalgam can be made?
PERCENT ALLOY BY WEIGHT
Amalgam
A
B
Silver
Tin
0.70
| 0.26
0.04
matrix 2&0.73\\ 5&0.27\\ 3&0.00 matrix ]
Copper
Click here to see answer by josgarithmetic(39659)

Question 1196138: Given P = ( 3 0 2
2 1 2
-3 -4 -6 )
a) if ( 2 6 n
-2 m 9
0 -2 1 ) is the cofactor of matrix P, find the values of m and without finding a new cofactor matrix.
b) find the adjoint matrix of P hence, find P-1 (identity P)

Click here to see answer by ElectricPavlov(122)

Question 1198615: The ABC Ink Company is a small family owned company that sells packages of ink cartridge refills
for smartpens. The Xavier set contains one blue ink refill and one black ink refill. The Yvonne set
includes two blue ink refills, three black ink refills, and one red ink refill. The Zena set includes
four blue ink refills, five black ink refills, and one red ink refill. The company has sold most of its
stock and has found that it has only 11 blue ink cartridge refills, 14 black ink cartridge refills, and 3
red ink cartridge refills. How many of each set should the company package to sell in order to use
all of the remaining ink cartridges so that there will be none left in inventory.

a. Describe what the variables represent: =

x=
y=
z=

b. Write the system of linear equations:

c. Solve the system of equations using Gauss-Jordan Elimination. Show all proper row operations
and the resulting matrices. Write the solution as an ordered triple, if appropriate.

d. Fill out the table to show the possible combinations of sets the company can package:
x y z

e. In at least one complete sentence with proper grammar and correct spelling, write the
solutions in terms of what the variables represent.
Click here to see answer by textot(100)

Question 1199304: Matrices
Suppose that three companies dominate the market for a certain product and are competing
against each other for a large share of the market. Suppose, further, that market surveys have
shown that buyers shift from one brand to another during a given year according to the
following schedule:
 C gains 20% of the customer of A and 40% remain to be the user of A.
 C gains 30% of its customer from B, retain 50% of its customer and loses 40% to B
 The customers of B have never shifted to A.
 It is found that the three companies have an equal share of the market today
a) Present the state vector and the transition matrix for the above case.
b) What share of the market will each possess two years from now?
c) Calculate the long run proportions that will be possessed by the three companies.
d) Company A wants to launch an advertising campaign that will cost Birr 200,000. Given the
result in (c) above, what would you advise them?

Click here to see answer by textot(100)

Question 1201037: 'n Beleggingsfirma bied drie aandele-opsies: Standard (x), Deluxe (y) en Gold (z) met 'n
aandeleverspreiding soos in die tabel aangedui. / An investment firm offers three stock
options: Standard (x), Deluxe (y) and Gold (z), with a stocks profile as indicated in the table.
10 30 40
15 10 30
10 20 30
Harmony
Billiton
Ashanti
H
B
A
x y z
Gestel die koste per aandeel vir Ashanti is R100, vir Billiton R150 en vir Harmony R175. Verder
is die inkomste per aandeel uit Ashanti R200, uit Billiton R250 en uit Harmony R270. 'n Kliënt
versoek 200 eenhede van Standard, 300 eenhede van Deluxe en 150 eenhede van Gold vir
'n persoonlike portefeulje. Stel 'n aandeleverspreidingmatriks A, kostematriks C,
inkomstematriks R en 'n eenhedematriks N saam. Pas matriksalgebra toe op hierdie matrikse
en bepaal die wins wat uit die portefeulje gemaak word. / Assume the cost per share for
Ashanti is R100, for Billiton R150 and for Harmony R175. The income per share from Ashanti
is R200, from Billiton R250 and from Harmony R270. A client requests 200 units of Standard,
300 units of Deluxe and 150 units of Gold for a personal portfolio. Compose a stocks profile
matrix A, a cost matrix C, an income matrix R and a unit matrix N. Apply matrix algebra to
these matrices and determine the profit made from the portfolio.
Click here to see answer by GingerAle(43)

Question 1204386: Let a ∈ R^n be a vector. Then {a} is linearly independent if and only if a does not equal 0.
Click here to see answer by ElectricPavlov(122)

Question 1205106: Let A and B be the following matrices:
A rotates vectors clockwise by a right angle. B projects vectors onto a line with direction (1; 4).
I don't know how to format matrices on here, so I will write the top row (horizontal) and the bottom row. This is a 2 x 2 matrix.
Calculate the matrix AB[-4 1][1 4].
Sorry if the matrices are unclear. In latex, you are calculating $\[\mathbf{A}\mathbf{B} \begin{pmatrix} -4 & 1 \\ 1 & 4 \end{pmatrix}.\]$
Click here to see answer by CPhill(2103)

Question 1205220: Determine the order of the following elements.
a) [1 2 3 4 5 2 3 4 5 1]
b) [1 2 3 4 4 3 1 2 ]

Click here to see answer by CPhill(2103)

Question 1209172: 2x+y=13 and 3x+2y=9
Click here to see answer by timofer(121)

Question 1208887: - x1 + 2 x2 + 3 x3 + x4 = 6
2 x1 + 3 x2 + x3 + 4 x4 = 8
2 x1 + 3 x2 + x3 + 3 x4 = 7
4 x1 - 2 x2 - 3 x3 + x4 = - 1
4 x1 - 5 x2 + 3 x3 - 2 x4 = 8
cramers rule

Click here to see answer by Alan3354(69443)

Question 1208735: let Z1 =a+bι ,and Z2 =c+ιd then find Z1.Z2
Click here to see answer by ikleyn(53213)

Question 1207305: Suppose a building contractor has accepted orders for five ranch style
houses, seven cape cod-style houses and twelve colonial-style houses.
These orders can be represented by the rowmatrix
Q = [5712]
Furthermore, suppose the raw materials that go into each type of house
are steel, wood, glass, paint, and labour. The entries in the matrix R below
give the number of units of each raw material going into each type of
house:
a)Compute the product, QR, the amount of each raw material
needed for the contract.
b)Suppose that steel costs N15/unit, wood costs N8/unit, glass, paint
and labour cost N5, N1, and N10 per unit respectively. These costs
are represented in the column matrix:
15
8
C = 5
10
Compute the product RC, that is, the cost of each type of house.
Click here to see answer by Edwin McCravy(20077)

Question 1207228: A Girls volleyball team is playing in a tournament. The probability that they will win their first match is 60%. The coach noticed that when the team wins a match, the probability that they will win the next match rises to 80%. If they lose a match, the probability that they will win the next match falls to 25%.
a) Write the initial probability matrix and the transition matrix.
b) Determine the probability that the team will win its third match.
Click here to see answer by ikleyn(53213)

Question 1207232: A volleyball team is playing in a tournament. The probability that they will win their first match is 60%. The coach noticed that when the team wins a match, the probability that they will win the next match rises to 80%. If they lose a match, the probability that they will win the next match falls to 25%.
a) Write the initial probability matrix and the transition matrix.
b) Determine the probability that the team will win its third match.
Click here to see answer by math_tutor2020(3826)

Question 1207042: so i have matrix A and B
and i want to find matrix X
where (A^-1)^TX=B
and the value for x=-1
3*3
A =
x-4 -6 -7
-3 x+1 10
0 -3 0

3*2
B=
1 3
3 0
0 -1
thanks for the help

Click here to see answer by math_tutor2020(3826)

Question 1207027: What is the determinant of the matrix F and G
F=
0 3 0 0 0
a 2 b c d
e 1 f g h
i 0 j k l
m 0 n o p
and
G =
a b c d
e-a f-b g-c h-d
i-m j-n k-o l-p
m n o p
if I have matrix
M=
a b c d
e f j h
i j k l
m n o p
that has a det=|M|=5
---------------------------------
so with matrix F I took 3 as common factor and multiplied it by |M|=5 which means |F|=15
and with G I just transform it to M which would mean |G|=|M|=5
is this correct?
any help is highly appreciated

Click here to see answer by math_tutor2020(3826)

Question 1206899: solve using gauss-jordan elimination
x+y+z=7
x-y+3z=21
5x+y+z=23
Click here to see answer by math_tutor2020(3826)

Question 1206899: solve using gauss-jordan elimination
x+y+z=7
x-y+3z=21
5x+y+z=23
Click here to see answer by ikleyn(53213)

Question 1206899: solve using gauss-jordan elimination
x+y+z=7
x-y+3z=21
5x+y+z=23
Click here to see answer by MathLover1(20850)

Question 1206243: Prove that a matrix A, 2x2 conmutes with any matrix 2x2 if and only if A conmutes with each of the matrices [1 0; 0 0], [0 1; 0 0], [0 0;1 0] [0 0; 0 1].
Find all those matrices A.
In this matrix notation a space refers to a change in column, and a semicolon to a change in row.
Thank you.
Click here to see answer by ikleyn(53213)

Question 1167786: A soap manufacturer decides to spend 60000 rupees on radio, magazine and tv advertising. if he spends as much on tv advertising as on magazines and radio together, and the amount spen on magazines and tv combined equals five times that spent on radio, what is the amount to spent on each type of advertising.using gauss elimination method?
Click here to see answer by greenestamps(13237)

Question 1206006: You as an executive of Rufus' communications recently traveled to London,
Paris, and Rome. You paid $280, $330, and $260 per night for loading in London,
Paris, and Rome, respectively, and your hotel bills totaled $4060. You spent $130,
$140, and $110 per day for meals in London, Paris, and Rome respectively, and
your expenses for meals totaled $180. If you spent as many days in London as you
did in Paris and Rome combined, how many days did you stay in each city? Use to
Gauss-Jordan method to solve
Click here to see answer by josgarithmetic(39659)

Older solutions: 1..45, 46..90, 91..135, 136..180, 181..225, 226..270, 271..315, 316..360, 361..405, 406..450, 451..495, 496..540, 541..585, 586..630, 631..675, 676..720, 721..765, 766..810, 811..855, 856..900, 901..945, 946..990, 991..1035, 1036..1080, 1081..1125, 1126..1170, 1171..1215, 1216..1260, 1261..1305, 1306..1350, 1351..1395, 1396..1440, 1441..1485, 1486..1530, 1531..1575, 1576..1620, 1621..1665, 1666..1710, 1711..1755, 1756..1800, 1801..1845, 1846..1890, 1891..1935, 1936..1980, 1981..2025, 2026..2070, 2071..2115, 2116..2160, 2161..2205, 2206..2250, 2251..2295, 2296..2340, 2341..2385, 2386..2430, 2431..2475, 2476..2520, 2521..2565, 2566..2610, 2611..2655, 2656..2700, 2701..2745, 2746..2790, 2791..2835, 2836..2880, 2881..2925, 2926..2970, 2971..3015, 3016..3060, 3061..3105, 3106..3150, 3151..3195, 3196..3240, 3241..3285, 3286..3330, 3331..3375, 3376..3420, 3421..3465, 3466..3510, 3511..3555, 3556..3600, 3601..3645