Lesson 12.2 Matrix Multiplication

3 Row and Column Order The rows in a matrix are usually indexed 1 to m from top to bottom. The columns are usually indexed 1 to n from left to right. Elements are indexed by row, then column.

If A is an m × n matrix and s is a scalar, then we let kA denote the matrix obtained by multiplying every element of A by k. This procedure is called scalar multiplication.

In matrix algebra, a real number is often called a SCALAR. To multiply a matrix by a scalar, you multiply each entry in the matrix by that scalar. Multiplying Matrices by a scalar

Scalar Multiplication - each element in a matrix is multiplied by a constant.

(-3) -5 -2(6)-2(-5) -2(3) Example PEMDAS – parenthesis first, do the matrix addition Do the scalar multiplication

The multiplication of matrices is easier shown than put into words. You multiply the rows of the first matrix with the columns of the second adding products Find AB Multiply across the first row and down the first column adding products. Put the answer in the first row, first column of the answer matrix.

Find AB We multiplied across first row and down first column so we put the answer in the first row, first column. Now we multiply across the first row and down the second column and we’ll put the answer in the first row, second column. Now we multiply across the second row and down the first column and we’ll put the answer in the second row, first column. Now we multiply across the second row and down the second column and we’ll put the answer in the second row, second column. Notice the sizes of A and B and the size of the product AB. Con’t

You can multiply two matrices A and B only if the number of columns of A is equal to the number of rows of B.

Examples: 2(3) + -1(5)2(-9) + -1(7)2(2) + -1(-6) 3(3) + 4(5) 3(-9) + 4(7)3(2) + 4(-6)

Dimensions: 2 x 3 2 x 2 *They don’t match so can’t be multiplied together.*

2 x 2 2 x 2 *Answer should be a 2 x 2 0(4) + (-1)(-2)0(-3) + (-1)(5) 1(4) + 0(-2) 1(-3) +0(5)