Matrix Entry or element Rows, columns Dimensions Matrix Addition/Subtraction Scalar Multiplication

In order to multiply matrices, the number of columns in the first matrix must match the number of rows in the second matrix. The other numbers represent the dimensions of the solution matrix. A x B = AB (2 x3) times (3 x 4) = (2x4) The first matrix is the row matrix. The second matrix is the column matrix. Circle the rows and columns before proceeding. The corresponding entries are multiplied and the results added to get a single entry of the product. Try examples on p. 376.

Tickets to the football game cost $2.50 for students, $5.00 for adults, and $4.00 for senior citizens. Attendance at the first game was 120 students, 185 adults, and 34 senior citizens. Attendance for the second game was 150 students, 210 adults, and 50 senior citizens. Use matrix multiplication to find the revenue for each game.

We call the number 1 the multiplicative identity because a∙1= 1∙a = a. What 2x2 matrix do you think we would call the identity matrix? Check your answer by multiplying it by the matrix below. Was your guess correct?

When two numbers multiply together to equal 1 we call them multiplicative inverses (or reciprocals). 2 ∙ 3 = When two matrices multiply together to equal an identity matrix we call them inverse matrices.

 To show that two matrices A and B are inverses, show that AB = BA = I.  But how do you find an inverse matrix?  Let A =  Find the inverse of A, often called A -1.

 Let A -1 =  Now A times A -1 must equal  Solve for a, b, c, and d.

 Answer:  Only square matrices have inverses. And not even all square matrices have an inverse.  A matrix which has an inverse is called invertible (or nonsingular). One which does not is called singular.

The previous method works for finding inverses for square matrices of any size. However, we so often find inverses of 2x2 matrices that we have developed a formula. For A =, A -1 = The number ad-bc is called the determinant of the 2x2 matrix. Therefore, if the determinant is 0, the inverse does not exist.

 The determinant of A = is noted as det(A) =

 3x3 matrices also have determinants. As with 2x2 matrices, the inverse of a 3x3 matrix only exists if the determinant is not 0. In this class we will use calculators to actually calculate 3x3 inverses.  The formula for the determinant of a 3x3 matrix is given on page 382 in your text.

 Consider the system x + 2y =8 3x + 4y = 6 Write a matrix equation in the form A∙X = B which represents this system. Now we want to solve for X. If A∙X = B, then A -1 ∙A ∙X = A -1 ∙ B and I X = A -1 ∙ B so X = A -1 ∙ B So you should be able to solve the above system by multiplying A -1 (which we already calculated) by B. Why would multiplying B by A -1 not work?