Period+6+Matrix+Arithmetic+and+Operations-Multiplication

Alexa McManus, Kyle McDonald, Alecia Christiano


 * Matrix multiplication** - a binary operation that takes a pair of matrices and produces another matrix.

A matrix can be multiplied by either a single number or another matrix. When multiplying a matrix, the final product of the multiplication will yield another matrix.

A matrix CANNOT be multiplied by another matrix if the number of columns in the 1st matrix does not equal the number of rows in the 2nd matrix!

Test you knowledge: Can you multiple theres matrices together?



Answer: YES!

Matrix A has four columns going from top to bottom and matrix B has four rows going from left to right. The number of COLUMNS in the first matrix ALWAYS has to equal the number of ROWS in the second matrix for there to be a solution. If they are not equal, the matrix is classified as undefined.


 * Note: Also, the number of ROWS in the first matrix and the number of COLUMNS in the second matrix gives you the dimensions of the product matrix. So, in the example above, Matrix A has two ROWS and Matrix B has three COLUMNS, so the product Matrix will have 2 rows and three columns (2x3).

__For example:__ You can multiply a matrix by a single number such a 3. The number 3 is called a scalar so this type of multiplication is called...

In order to multiply a matrix by a single number you must multiply each number in the matrix by the scalar. As you multiply the numbers inside the matrix by the scalar, start creating the product of the scalar and the matrix by creating a new matrix with the products of each multiplication problem in the same order as the they were formed in the original matrix.
 * Scalar multiplication**


 * Multiplying a Matrix by Another Matrix**

Use the "Dot Product"- multiply matching members, then sum them up. 1. Multiply the first row by the first column. 2. Mulitply the first row by the second column. 3. Multiply the second row by the first column. 4. Multiply the second row and second column. (1, 2, 3) • (7, 9, 11) = 1×7 + 2×9 + 3×11 = 58 (1, 2, 3) • (8, 10, 12) = 1×8 + 2×10 + 3×12 = 64(4, 5, 6) • (7, 9, 11) = 4×7 + 5×9 + 6×11 = 139 (4, 5, 6) • (8, 10, 12) = 4×8 + 5×10 + 6×12 = 154



__Example 2:__

Find the Product AB for the following matrices:



In order to calculate AB, it is best to write them side by side like this:



Now you have to multiply the **ROWS** of A by the **COLUMNS** of B. What this means is you have to multiply the first number in the first row of A by the first number in the first column of B, the second number in the first row of A by the second number in the first column of B, etc. When you are done multiplying a row by a column, you add all of these products up to get an entry in matrix AB.

Here is an illustration that shows the order in which you multiply the rows of A by the columns of B, and how the sums should be entered in the matrix AB (Your final answer):

Example 3- Harder Multiplying Matrix Problem Evaluate the given expression: 1. Take the product of the matrices in the first term of the expression. 2. Take the product of the matrices in the second term of the expression. 3. Add the two resulting matrices.

If you still need help with multiplying matrices, here is a link to a video that offers an explanation to the entire process. http://www.khanacademy.org/math/algebra/algebra-matrices/v/matrix-multiplication--part-1

**Web Bibliography** 1.) "How to Multiply Matrices." www.mathisfun.com. MathIsFun.com, 2011. Web. 2 Apr. 2012. .

2.) "Matrix Multiplication." www.mathwarehouse.com. mathwarehouse.com, n.d. Web. 2Apr. 2012. 

3.) "Scalar and Matrix Multiplication." //http://www.purplemath.com/modules/mtrxmult.htm .// Purplemath.com, n.d. Web. 6 Apr. 2012. < http://www.purplemath.com/modules/mtrxmult. htm >.