Period+8-Matrix+Arithmetic+and+Operations-Multiplication

Paige and Molly

As you learned in Scalar Multiplication, you can multiply matrices by a number, but you can also multiply a matrix by another matrix!
 * Matrix Multiplication**

__ How to Multiply Two Matrices __ When multiplying two matrices, write them side by side. Then, multiply the //rows// of the first matrix by the //columns// of the second one. Multiply each entry in the first row of the first matrix with the corresponding entry in the first column of the second matrix, and add these products together. Then, put this new number in the new matrix. If you are multiplying the farthest up row by the farthest right column, the new number goes in the upper right corner of the new matrix. Likewise, if you are multiplying the farthest up row by the farthest left column, the new number goes in the upper left corner of the new matrix, and so forth.

If Matrix A has //m// rows and //n// columns, and Matrix B has //n// rows and //p// columns, then their product AB has //m// rows and //p// columns. Because of this, AB ≠ BA. This is shown by the following example:



BUT

Another example that AB ≠ BA:

NOTE that a matrix CANNOT be multiplied by another matrix if the number of columns in the first matrix does not equal the number of rows in the 2nd matrix!

__The Identity Matrix__
 * CHARACTERISTICS:**
 * It is a square matrix (having the same number of rows and columns)
 * It has 1's on the diagonal, and 0s everywhere else.
 * It's symbol is capital "I"
 * When you multiply it, the original is unchanged:
 * A*I=A
 * I*A=A

if
 * Example of Multiplying by the Identity:**

Then, multiplying A by a 3X3 identity matrix, We find out that A remains the same!

Matrix multiplication is show by this formula: where 1 ≤ //i// ≤ //m// and 1 ≤ //j// ≤ //p//.

1. AB = ?
 * Practice Problems**

2. BC = ?

3. CB = ?


 * Answers**

1. Cannot be solved, because the number of columns in A is not equal to the number of rows in B.

2. 3. Websites used: http://www.mathsisfun.com/algebra/matrix-multiplying.html http://en.wikipedia.org/wiki/Matrix_multiplication http://www.purplemath.com/modules/mtrxmult.htm http://www.algebralab.org/lessons/lesson.aspx?file=algebra_matrices_multiplying.xml Step-by-Step Practice Problems: http://www.mathwarehouse.com/algebra/matrix/multiply-matrix.php