Period+6+-+Determinants

__**Determinants**__ Sara Petro, Sophie Du, Hana Chung Honors Pre-Calc, Period 6

__**Intro to Determinants:**__
 * The determinant is a real number associated with each matrix.
 * Only square matrices can have determinants.
 * A square matrix has the same number of rows as columns.
 * Ex: 2x2, 3x3, 20x20 (meaning mxm)
 * Notation for determinants: Both lAl and det (A) mean the determinant of A.

__**a. Determinant of a 2x2**__

A 2x2 matrix is written as you see below. A equals a and b over c and d, which are encased in parentheses. Keep in mind that when writing the matrix as a determinant. Absolute value symbols are put around A and the parentheses are changed to absolute value signs. Note: Determinants and absolute values are similar, but they are not identical. Determinants CAN be negative. � >> >> Example 1: of Girls || Number of Boys || Example 2:
 * How to multiply a 2x2 matrix
 * [[image:http://www.onlinemathlearning.com/image-files/matrices002_0007.gif]]
 * *Remember: lAl means the determinant of A.
 * *Essentially, the determinant of the 2x2 is found by subtracting the product of the diagonals.
 * *In other words, cross multiply //a & d// and //b & c// and subtract the product of //bc// from //ad.//
 * Data is summarized sometimes by using tables. When the tables have two columns of information, it can be written as a 2 x 2 matrix.
 * For example:
 * || Number
 * ClassA || 8 || 7 ||
 * ClassB || 5 || 4 ||
 * *This chart can be written as a 2x2 matrix, which can be multiplied by using the method mentioned above
 * When it is written as a matrix, the chart looks the exact same but without the categories and brackets are placed around the numbers, as seen above the example.
 * Step 1: Find the determinant of A if a = 8, b = 7, c = 4, d = 4 (look at how to multiply 2x2 matrix above)
 * Step 2: lAl = (8*4) - (7*5)
 * *Answer: lAl = -3
 * This example is not written in a table, it is written strictly as a matrix.
 * Using the steps outlined above, find the determinant of A, or lAl

If you had trouble with that... lAl = (2*3) - (-1*1) = (6) - (-1) = (6) + (1) = 7
 * The answer to this problem is 7.

If you are still having problems understanding 2x2 determinants, take a look at [|Math Centre 2x2 Determinants]

2x2 Matrices are not just for math, they are used in real life! They are used to organize data as shown above, but they are also used in the business and marketing world!


 * b. __Determinant of a 3x3__**
 * Determinant of a 3x3 matrix is called third-order determinant.
 * Expansion by diagonals is one method of evaluating a 3x3 matrix.

Let:
 * Example 1:**


 * Remember: Both lAl and det (A) mean the determinant of A.

__Step 1:__ Take the first two columns and write them over to the right of the matrix in the same order. __Step 2:__ Start in the upper left hand corner and multiply the variables in the diagonal in green. Then multiply the variables in the purple diagonal. Multiple the variables in the blue diagonal. Add are three together.

= (xbt) + (ycr) + (zas)
__Step 3:__ Next, do the same again but start from the upper right hand corner. Multiply the variables in the orange diagonal. Multiply the variables in the brown diagonal. Multiply the variables in the pink diagonal. Add all three together.

= (yat) + (xcs) + (xbr)
__Step 4:__ Subtract the answer from step 2 by the answer from step 3. This is the final answer to the determinant of A.

*Answer= lAl=xbt+ycr+zas-yat-xcs-zbr
Let:
 * Example 2**

__Step 1:__ Take the first two columns and write them over to the right of the matrix in the same order

__Step 2:__ Start in the upper left hand corner and multiply the variables in the diagonal in green. Then multiply the variables in the purple diagonal. Multiple the variables in the blue diagonal. Add are three together.

=38
__Step 3:__ Next, do the same again but start from the upper right hand corner. Multiply the variables in the orange diagonal. Multiply the variables in the brown diagonal. Multiply the variables in the pink diagonal. Add all three together.



__Step 4:__ Subtract the answer from step 2 by the answer from step 3. This is the final answer to the determinant of A.

lAl=35 *Answer
__Now you try example 3!!__



*Answer= lAl=55
If you're still having troubles, take a look at these references: [] [] -This one has a great video. Click on the first one.

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 * __Web Bibliography__**