Determinants

Connor Metcalf Solving Determenants.

__**Examples**__ 1) A determinant is a special kind of matrix, one that has a numerical value. The value of a matrix may be 3, for example. We write a determinant with vertical lines around it: code   |  2  1 |    | -3  4 | = 8-(-3) = 11 code Here you see four vertical lines, because I am limited in my choice of symbols here. Normally, draw just one longer line on each side of the determinant. The above 2x2 determinant equals 11. This is how you evaluate a 2x2 determinant: code    | a  b |    | c  d | = ad-bc code Let's solve our first system of equations (in two variables), rewriting it as: code    3x-y=1    2x-y=-3 code This gives us three determinants, and these two equations code         | 1 -1 |        | 3  1 |         |-3 -1 |        | 2 -3 |     x= __________   y= __________         | 3 -1 |        | 3 -1 |         | 2 -1 |        | 2 -1 | code Remember that determinants are numbers. The above equations are just numbers divided by numbers. The denominators are determinants derived from the left side of our two equations (the four coefficients). If the denominator=0, then the system has no solution. You already knew that you can't divide by zero. The two numerators are made up of the denominator determinant, with the two numbers from the right side of our equations inserted into it. We can evaluate these three determinants, and we get x=(-1-3)/(-3+2)=4 and y=(-9-2)/(-3+2)=11. That is the answer that we got above.

Determinants may seem like a waste of time with two equations and two unknowns. But you can get good at using them. Besides, determinants work well when programmed into a computer. 3x3 determinants work much the same way, and are not much more difficult. We begin to see how useful determinants can be, when trying to solve three or more equations. Let's solve our system of three equations: code x-2y+ z=-1 2x+ y-3z=3 3x+3y-2z=10 code We construct our determinants in the same way: code |-1 -2 1 |        | 1 -1  1 |        | 1 -2  1 |         | 3  1 -3 |        | 2  3 -3 |        | 2  1  3 |         |10  3 -2 |        | 3 10 -2 |        | 3  3 10 |     x= _____________   y= _____________   z= _____________ | 1 -2 1 |        | 1 -2  1 |        | 1 -2  1 |         | 2  1 -3 |        | 2  1 -3 |        | 2  1 -3 |         | 3  3 -2 |        | 3  3 -2 |        | 3  3 -2 | code There are two ways to evaluate a 3x3 determinant. I will show you both ways. Here is a general 3x3 determinant: code | a b  c | | d e  f | = aei+bfg+cdh-afh-bdi-ceg | g h  i | code Do you see the pattern there? We go along diagonals, the last three being negative. This method can be easily extended to 4x4 and larger determinants. And it can be programmed into a computer. This is the method that I learned. But the following method seems to be more common: code | a b  c |     | e f |     | d f |     | d e | | d e  f | = a | h i | - b | g i | + c | g h | | g h  i | code or code | a b  c |     | e f |     | b c |     | b c | | d e  f | = a | h i | - d | h i | + g | e f | | g h  i | code We just make our 3x3 determinant into the sum of three 2x2 determinants. Again there is an easy pattern here. We have **a** times the 2x2 determinant made up of numbers not from the same rows and colums of **a**. We can choose any row or column to form our three determinants. If we had chosen the second column, instead of the first, the signs would have been -+- instead of +-+. This method can also be programmed into a computer. Let's solve our three equations: code |-1 -2 1 |        | 1 -1  1 |        | 1 -2 -1 |         | 3  1 -3 |        | 2  3 -3 |        | 2  1  3 |         |10  3 -2 |        | 3 10 -2 |        | 3  3 10 |     x= _____________   y= _____________   z= _____________ | 1 -2 1 |        | 1 -2  1 |        | 1 -2  1 |         | 2  1 -3 |        | 2  1 -3 |        | 2  1 -3 |         | 3  3 -2 |        | 3  3 -2 |        | 3  3 -2 | code x=(2+60+9-9-12-10)/(-2+18+6+9-8-3)=40/20=2 and y=(-6+9+20+30-4-9)/20=40/20=2 and z=(10-18-6-9+40+3)/20=20/20=1. I will substitute those numbers back in to the three equations (we have to do this, as you should always check for mistakes) to see if x=2, y=2, z=1 is the solution. It worked. That is the solution. This may all seem difficult. But, by systematizing the process, it actually makes solving systems of equations easy.
 * 3x3 determinants:**

Incidentally, here is a formula for the area of a triangle, given the rectangular coordinates of the vertices: code | x1 y1  1 | A = 1/2 | x2 y2  1 | | x3 y3  1 | code A more complicated formula, involving three determinants, gives the area of a triangle, given the rectangular coordinates of the vertices in three-space.

("Jim Loy." __Determinants__. 2001. 5 Nov. 2008 [|www.jimloy.com/algebra/determin.htm].)

You would start by multiplying diagonally you would start with 2 * 6 which would equal 12. Then you would multiply -1 * 5 which would equal -5. After you are done multiplying its time to add them all together. You would add 12 + 5 since 5 was a negative you automatically switch the signs whenever you have a negative to a positive but if you are given a positive you will be adding a negative. So the answer would be 17. 3)** You would rewrite the first 2 columns on the right which would look like this: Then you would diagonally multiply the same way you would in the first 2 examples and it would look like this 0 * 3* 1=0, 0 * 1 * 4=0, 0 * 2 * 4= 0 and the entire problem would equal 0.
 * 2)
 * 2 5 |
 * -1 6 |
 * 3 1 0|
 * 1 4 0|
 * 2 4 0|
 * 0 3 1 |
 * 0 1 4 |
 * 0 2 4|

**__Problems__**

1) 2) 3) 4) 5)
 * 1 1|
 * 2 2|
 * 5 6|
 * 3 0|
 * 2 8|
 * 4 6|
 * 3 5 6|
 * 8 2 4|
 * 5 8 0|
 * 3 5 7|

__**Picture of Solving Determinants**__ ("Algebra 2. Lesson 1." __ETAP__. 2007. 5 Nov. 2008 http://images.google.com/imgres?imgurl=http://www.etap.org/demo/algebra2/image52.gif&imgrefurl=http://www.etap.org/demo/algebra2/instructiontutor_last.html&h=648&w=516&sz=4&hl=en&start=6&usg=__asubl839qv-icvqofin_08dcsek=&tbnid=n0c22rdna7abym:&tbnh=137&tb.)

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 * __Definitions__

** Entries (Or Elements)- A 1, B 1 , A 2 and B 2 are the entries Determinant of coefficients- D is called the determinant of coefficients Cramers Rule- The solution of a linear system in determinant form (4) is called Determinant- a square array of numerals set off with vertical bars, which names a real number. The numerals in the array are called the entries or elements of the determinant. The order of determinant is the number of rows or columns. Rows- 2 horizontal lines of numbers Columns- 2 vertical lines of numbers Order 2- The plane 4 numbered determinant is called order 2 or second-order determinant Minor- an element in a determinant is the determinant resulting from the deletion of the row and column containg the element

1) choose a row or column and form the product of each element in the row or column with its minor 2) use the product obtained or its negative according as the sum of the number of the row and number of the column containing the element is even or odd 3) the sum of the resulting numbers is the value of the determinant
 * Way to solve**