Chapter Seven Linear Systems and Matrices

Warm-up #1 The University of Georgia and Florida State University scored a total of 39 points during the 2003 Sugar Bowl. The points came from a total of 11 different scoring plays, which were a combination of touchdowns, extra point kicks, and field goals, worth 6, 1, and 3 points respectively. The same number of touchdowns and field goals were scored. How many touchdowns, extra point kicks, and field goals were scored during the game?

HWQ #2 Write the partial fraction decomposition:

7.5 – Operations with Matrices Equality of Matrices Matrix Addition and Scalar Multiplication Matrix Multiplication Applications

7.5 – Equality of Matrices A matrix is equal to another if the dimensions are the same and the entries are all equivilant. Ex: Solve for x and y:

7.5 – Addition of Matrices Example: Matrix addition is a piece-wise addition and therefore the dimensions need to be exactly the same. Ex: Find A+B

7 Matrix Addition To add matrices: 1. Check to see if the matrices have the same order. 2. Add corresponding entries. Example: Find the sums A + B and B + C. A has order 3 2 and B has order 2 3. So they cannot be added. C has order 2 3 and can be added to B.

8 Matrix Subtraction To subtract matrices: 1. Check to see if the matrices have the same order. 2. Subtract corresponding entries. Example: Find the differences A – B and B – C. Since B is of order 2 2 and C is of order 3 2, they cannot be subtracted. A and B are both of order 2 2 and can be subtracted.

9 Scalar Multiplication Example: Find 2A and –3A for A =. If A is an m n matrix and C is a scalar, then the m n matrix CA is the scalar multiple of A by C.

10 Example: Matrix Operations Example: Calculate the value of 3A – 2B + C with

7.5 – Matrix Equations With matrix equations, the variable you are solving for is a matrix. Ex: Solve the matrix equation 3X+A=B given matrices A and B:

7.5 – Matrix Multiplication 2 matrices have a product if the # of columns of the left matrix = the # of rows of the right matrix. Multiply across on the left, down on the right. Ex: Find the product AB, then try BA.

7.5 – The Identity Matrix This is a square matrix in which all of the diagonal entries are ones and all of the off-diagonal entries are zero. Ex: Multiply matrix A by the identity matrix I.

7.5 – Application Find the equation of the parabola that passes through the points.

7.5 – Application An inheritance of $20,000 is divided among 3 investments yielding $1780 in interest per year. The interest rates for the three investments are 7%, 9%, and 11%. Find the amount of each if the amount invested at 7% was $2000 less than half of the total investment.

Homework 7.4 pg. 501: 71, pg odd, 15, odd, 65,67