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From Wikipedia

How to Solve It

How to Solve It (1945) is a small volume by mathematician George Pólya describing methods of problem solving.

Four principles

How to Solve It suggests the following steps when solving a mathematical problem:

  1. First, you have to understand the problem.
  2. After understanding, then make a plan.
  3. Carry out the plan.
  4. Look backon your work.How could it be better?

If this technique fails, Pólya advises: "If you can't solve a problem, then there is an easier problem you can solve: find it." Or: "If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?"

First principle: Understand the problem

"Understand the problem" is often neglected as being obvious and is not even mentioned in many mathematics classes. Yet students are often stymied in their efforts to solve it, simply because they don't understand it fully, or even in part. In order to remedy this oversight, Pólya taught teachers how to prompt each student with appropriate questions, depending on the situation, such as:

  • What are you asked to find or show?
  • Can you restate the problem in your own words?
  • Can you think of a picture or a diagram that might help you understand the problem?
  • Is there enough information to enable you to find a solution?
  • Do you understand all the words used in stating the problem?
  • Do you need to ask a question to get the answer?

The teacher is to select the question with the appropriate level of difficulty for each student to ascertain if each student understands at their own level, moving up or down the list to prompt each student, until each one can respond with something constructive.

Second principle: Devise a plan

Pólya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:

  • Guess and check
  • Make an orderly list
  • Eliminate possibilities
  • Use symmetry
  • Consider special cases
  • Use direct reasoning
  • Solve an equation

Also suggested:

  • Look for a pattern
  • Draw a picture
  • Solve a simpler problem
  • Use a model
  • Work backward
  • Use a formula
  • Be creative
  • Use your head/noggin

Third principle: Carry out the plan

This step is usually easier than devising the plan. In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work discard it and choose another. Don't be misled; this is how mathematics is done, even by professionals.

Fourth principle: Review/extend

Pólya mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked and what didn't. Doing this will enable you to predict what strategy to use to solve future problems, if these relate to the original problem.

The book contains a dictionary-style set of heuristics, many of which have to do with generating a more accessible problem. For example:

The technique "have I used everything" is perhaps most applicable to formal educational examinations (e.g., n men digging m ditches) problems.

The book has achieved "classic" status because of its considerable influence (see the next section).

Other books on problem solving are often related to more creative and less concrete techniques. See lateral thinking, mind mapping, brainstorming, and creative problem solving.

Influence

  • It has been translated into several languages and has sold over a million copies, and has been continuously in print since its first publication.
  • Marvin Minsky said in his influential paper Steps Toward Artificial Intelligence that "everyone should know the work of George Pólya on how to solve problems."
  • Pólya's book has had a large influence on mathematics textbooks as evidenced by the bibliographies for mathematics education.
  • Russian physicistZhores I. Alfyorov, (Nobel laureate in 2000) praised it, saying he was very pleased with Pólya's famous book.
  • Russian inventor Genrich Altshuller developed an elaborate set of methods for problem solving known as TRIZ, which in many aspects reproduces or parallels Pólya's work.


From Yahoo Answers

Question:Hey guys. Having some serious problem solving these: (1) A historian knows there were 29 men on a ship. The daily wine allowance was 252 ounces, and the monthly payroll as 30,000 maravedis. The historian still assumes the daily wine allowance per man was 8 ounces for seamen and 12 ounces for officers. The salary was 1000 maravedis a month for a seamen and 2000 for an officer. Calculate how many seamen and officers made the voyage, or show that the historians assumptions are incorrect because the equations are inconsistent. (2) On a European vacation a student spent $30 a day for housing in England, $20 a day in France, and $20 a day in Spain. For food she spent $20 a day in England, $30 a day in France, and $20 a day in Spain. She also spent $10 a day in each country for incidental expenses. Her records show a total of $340 spent for housing, $320 for food, and $140 for incidental expenses. Calculate the number of days the traveler spent in each country or show that the records must be incorrect because the amounts spent are incompatible with each other. PLEASE help me! I haven't a clue as to how to start organizing these at all! Please do NOT direct me to a "math problem solver" website as it is not at all helpful. If you know how to solve these personally, please help me out. It is TRULY appreciated! Thank you.

Answers:S+O=29 S*ws+O*wo = 252 S*ps +O*po=30,000 Ican only write 3 equations for 6 unknowns so I'd better hope the historian's information is inconsistent S+O=29 8S+12O = 252 1000S+2000O=30,000 Quickly solving the first two S=24 O =5 On solving the second 2 S=36 O=-3 On solving 1 and 3 and ignoring 2 S=28 O=1 Obviously inconsistent (2) Same idea 30*dE+20*dF+20*dS =340 10*(dE+dF+dS)=140 20*dE+30*dF+20*dS=320 3 equations 3 unknowns, there is a unique solution Write as matrix |30 20 20| |deE| |340| |10 10 10| |dF | = |140| |20 30 20| |dS | |320| Solve dE =6 dF =4 dS =4 Hope that helps

Question:I have a 3x3 matrix and a 3x1 how do I solve for the missing elements? A=1 1 -1 B= C = 2 2 0 4 2 3 -1 9 2 How do I go about solving this problem?

Answers:I worked out your problem using a math program. The result is a PDF file stored here... https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B93wLOqLZwn_NmI1NGY4OWUtYjNlNi00NzJiLTg2OWMtNTQxYmU5NmJkOGQx&hl=en&authkey=CIXQybgL Hope this helps

Question:Okay, once again I need help on how to set up an equation for an algebra word problem. Here it is: Hilary has three times as much money as Paul. Jeff has $4 less than Hilary and $5 more than Paul. How much money does each have? Thanks!

Answers:HINT: Write what you know. Let h = Hilary's money Let p = Paul's money Let j = Jeff's money ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ METHOD: Elimination/Substitution ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Given: Hilary has three times as much money as Paul. Means: h = 3p Given: Jeff has $4 less than Hilary Means: j = h - 4 Given: Jeff has $5 more than Paul. Means: j = p + 5 You have 3 equations. h = 3p j = h - 4 j = p + 5 Substitute h with 3p in the 2nd equation. j = h - 4 j = 3p - 4 Look at this equation and the 3rd equation. j = 3p - 4 j = p + 5 Multiply the 2nd of these equations by -1. -1(j) = -1(p + 5) -j = -1(p) - 1(5) -j = -p - 5 Add this to the other equation. j = 3p - 4 -j = -p - 5 ---------------- 0 = 2p - 9 9 = 2p 9 / 2 = p 4.5 = p Plug this into h = 3p to find h. h = 3p h = 3(4.5) h = 13.5 Plug this into j = h - 4 to find j. j = h - 4 j = 13.5 - 4 j = 9.5 ANSWER: Hilary has $13.50, Paul has $4.50, and Jeff has $9.50. CHECK: h = 3p 13.5 = 3(4.50)? 13.5 = 13.5? TRUE j = h - 4 9.5 = 13.5 - 4? 9.5 = 9.5? TRUE j = p + 5 9.5 = 4.5 + 5? 9.5 = 9.5? TRUE ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ METHOD: Cramer's rule & matrices ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ h = 3p ==> h - 3p + 0j = 0 j = h - 4 ==> h + 0p - j = 4 j = p + 5 ==> 0h - p + j = 5 The matrix for D is: ......[1 -3 0] D = [1 0 -1] ......[0 -1 1] Repeat the first two columns and evaluate for D. ......[1 -3 0 1 -3] D = [1 0 -1 1 0] ......[0 -1 1 0 -1] D = (1)(0)(1) + (-3)(-1)(0) + (0)(1)(-1) - (0)(0)(0) - (-1)(-1)(1) - (1)(1)(-3) = 0 + 0 + 0 - 0 - 1 - -3 = 2 The matrix for Dh is: ........[0 -3 0] Dh = [4 0 -1] ........[5 -1 1] Repeat the first two columns and evaluate for Dx. ........[0 -3 0 0 -3] Dh = [4 0 -1 4 0] ........[5 -1 1 5 -1] Dh = (0)(0)(1) + (-3)(-1)(5) + (0)(4)(-1) - (5)(0)(0) - (-1)(-1)(0) - (1)(4)(-3) = 0 + 15 + 0 - 0 - 0 - -12 = 27 The matrix for Dp is: ........[1 0 0] Dp = [1 4 -1] ........[0 5 1] Repeat the first two columns anDy evaluate for Dy. ........[1 0 0 1 0] Dp = [1 4 -1 1 4] ........[0 5 1 0 5] Dp = (1)(4)(1) + (0)(-1)(0) + (0)(1)(5) - (0)(4)(0) - (5)(-1)(1) - (1)(1)(0) = 4 + 0 + 0 - 0 - -5 - 0 = 9 The matrix for Dj is: ........[1 -3 0] Dj = [1 0 4] ........[0 -1 5] Repeat the first two columns anDz evaluate for Dz. ........[1 -3 0 1 -3] Dj = [1 0 4 1 0] ........[0 -1 5 0 -1] Dj = (1)(0)(5) + (-3)(4)(0) + (0)(1)(-1) - (0)(0)(0) - (-1)(4)(1) - (5)(1)(-3) = 0 + 0 + 0 - 0 - -4 - -15 = 19 Evalute for h, p, and j. h = Dh / D = 27 / 2 = 13.5 p = Dp / D = 9 / 2 = 4.5 j = Dj / D = 19 / 2 = 9.5 (13.5, 4.5, 9.5) ANSWER: Hilary has $13.50, Paul has $4.50, and Jeff has $9.50.

Question:Hi, I'm having trouble solving this problem: We're given that that X(transpose) * A * X = (3x^2-2xy+y^2) We're also given that X is a 1x2 matrix with the entries x,y. We're also given that A is a 2x2 symmetric matrix. Could anyone help me find solve to find A? Thanks!

Answers:No cause what you have given is an impossible situation. if x is a 1x2 matrix, its transpose is a 2x1 matrix... you cant multiply that by a 2x2 matrix.... ima assume you meant its a 2x1 matrix. This takes a long time so im just gunna tell you how to do it. (x y) {a1 a2 ; a3 a4} {x ;y} = (3x^2-2xy+y^2) a ; indicates a new line. Multiply it out and solve for all the a's

From Youtube

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