Chapter 4

1. Chapter 4 Matrices By: Matt Raimondi

2. 4-1 and 4-2 Introduction to Matrices • Matrices used when a group of numbers or variables are blocked together. • Each entry of a matrix is referred to as an element. • The dimension of a matrix is defined by the number of rows by the number of columns. So a 3 x 4 matrix would have 3 rows and 4 columns. • When a matrix is being multiplied by a scalar, any real number, it is distributed to each element of the matrix. • Matrices may only be added or subtracted if they have the same dimensions. Then each element is added or subtracted to the corresponding element of the other matrix.

3. Here’s an example of scalar multiplication. Each element was multiplied by 4. Here’s an example of matrix addition. Look how each element of A is simply added to the corresponding element of matrix B. The only reason these two matrices can be added is because they have the same dimension. Both are 3 x 2. A A B B A+B= 4-1 and 4-2 Examples 4A -4+2 -7 + (-5) -9 + 7 -9+x 5+(-y) -2+2 A+B

4. 4-1 and 4-2 Introduction to Matrices • Two matrices are equal if and only if the following are true: • They have the same dimensions. • All of the elements of the first matrix are equal to the corresponding elements of the second. Lets look at some examples:  = 

5. = 4-1 and 4-2 Solving Matrix Equations • The only way two matrices can be equal is if each element of the corresponding matrix is equal. Let’s take a look at an example problem: To see if this is true we have to solve for x. If all of the statements are true then the matrices are equal. Since the 2nd row 1st column element says x=7 and the 1st row 2nd column element is 3x=21, the matrices are equal. • 3x 5 21 • X -6 7 -6 5 = 5 3x=21 X=7 -6=-6

6. 4-1 and 4-2 Problems • Solve the following matrix equations with the following matrices A, B, and C: • -3A 2) B - A 3) 2B • 4) B+A 5) A + 2B 6) 6B + 3A • 7) Solve A=C for x and y. • Answers on next page.

7. 4-1 and 4-2 Answers 1) 2) 3) 4) 5) 6) • x=8 y=5

8. 4-3 Multiplying Matrices • Unlike adding and subtracting matrices, they do not have to have the same dimensions to be multiplied. • The number of columns in the first matrix must match the number of rows of the second matrix. • The product will have the number of rows of the first matrix and the number of columns of the second. • For example, a 2x3 matrix multiplied by a 3x5 matrix will produce a 2x5 matrix. • If the number of columns of the first matrix does not match the number of rows of the second, the product does not exist. • To find the product, the rows of the first matrix are multiplied by the columns of the second.

9. 0(4)+3(0) 0(3)+3(3) 0(1) +3(2) -1(4)+1(0) -1(3)+1(3) -1(1)+1(2) Lets take a look at the product of these two matrices. The first is a 2x2 and the second is a 2x3 matrix. Since the number of columns of the first matrix matches the number of rows of the second, the product exists. Observe how each row of the first matrix is being multiplied by the columns of the second. The product will be a 2x3 matrix. 4-3 Examples

10. 4-3 Problems Given the dimensions of the matrices, state the size of the product if it exists. 1) 4x8 * 8x2 2) 3x3 * 3x2 3) 3x2 * 3x3 4) 1x4 * 4x1 5) 4x1 * 4x1 6) 7x5 * 5x1 Compute the following products. 7) 8) 9) Answers on next page

11. 4-3 Answers 1) 4x2 2) 3x2 3) Non existent 4) 1x1 5) Non existent 6) 7x1 7) 8) 9)

12. 4-4 Determinants • All square matrices have a determinant. The only ones we will deal with are 2x2 and 3x3 matrices. • To find the determinant of a 2x2 matrix, you use the rule for second order determinants. • ad-bc • There are two ways to find the determinant of a 3x3 matrix. • First is expansion by minors. To get your minors, pick any row of the matrix look at the first digit, it will become a multiple for a 2x2 matrix. Ignore the other numbers in the row and cover up the other numbers in the column. You will only have 4 numbers left in the matrix and take that as a 2x2 matrix. Use the digit you started with as a multiple of the matrix. Repeat those steps for the remaining two digits in the row. Look at the next page for the formula.

13. 4-4 Determinants • This is assuming you pick the first row for your multiples but remember that you can pick any row. = a* -b* +c* • Observe how when a is the multiple, the corresponding matrix is only the elements left over when the row and column of a are covered. • The other way to find the determinant of a 3x3 matrix is by using diagonals. Look at the next page to see how it works.

14. First take the first two columns and add them on to the end. Then draw diagonals from the first entry of each row down and to the right. You obtain aef, bfg, and cdh. Then start at the bottom and draw diagonals up and to the right. You get gec, hfa, and idb. The determinant will equal aef+bfg+cdh-gec-hfa-idb. 4-4 Determinants

15. 4-4 Problems Solve the following: 1) 2) 3) Solve by expansion by minors: 4) 5) 6) Solve by diagonals: 7) 8) 9) Answers: 1) 14 2)-22 3)-22 4)-32 5)-95 6) 369 7)-323 8)-30 9) 90

16. 4-5 Identity Matrix • Similarly to multiplication of real numbers, matrices have inverses and identities. • We know that any number times 1 will give us the original number. Likewise, any square matrix times the identity matrix of equal size equals the original matrix. * =

17. 4-5 Inverse of a Matrix • In multiplication (1/a)*a = 1. This is the same as (a-1)a = 1. The inverse of a number times itself will equal 1. But with matrices, it will equal the identity matrix. A= A-1 = A* A-1= A-1*A= I2 =

18. 4-6 Using Matrices to Solve Systems of Equations • A system of equations such as: 5x - 9y = -20 -4x + 3y = -5 can be solved using matrices. • First, make a matrix of the coefficients. • Then make a matrix of the variables and one for the solutions. * =

19. 4-6 Using Matrices to Solve Systems of Equations • To solve for x and y we need to get rid of the coefficient matrix. • We can do this by multiplying by the inverse matrix on both sides A = A-1 =