4.2 An Introduction to Matrices

1. 4.2 An Introduction to Matrices Algebra 2 Mrs. Spitz Fall 2006

2. Objectives • Create a matrix and name it using its dimensions • Perform scalar multiplication on a matrix • Add matrices • Find unknown values in equal matrices.

3. Assignment: • Pgs. 164-165 #11-25 all

4. What is a matrix? • It isn’t the movie. • A system of rows and columns • A problem-solving tool that organizes numbers or data so that each position in the matrix has a purpose.

5. Connection • In Algebra, a matrix is not expressed as a table, but as an array of values. Each value is called an element of the matrix. Suppose we want to write the coordinates of the vertices of the ABC as a matrix.

6. Connection • Suppose we want to write the coordinates of the vertices of the ABC as a matrix. Let row 1 be the x-coordinates. Let row 2 be the y-coordinates. Each column represents the coordinates of vertices A, B, and C. 3 columns 2 rows

7. How is it named? • Usually named using an uppercase letter. We might call the matrix for ABC matrix T to stand for triangle. A matrix can also be named by using the matrix dimensions with the letter name. The dimensions tell how many rows and columns there are in the matrix. The matrix on the preceding slide would be named T2x3 since it has two rows and 3 columns. Note: Please realize that T2x3 does not have the same dimensions as T3x2 .

8. What if they don’t have the same number of columns or rows? What if they do? • Certain matrices have special names. A matrix that has only one row is called a row matrix, and a matrix that has only one column is called a column matrix. A matrix that has the same number of rows and columns is called a square matrix. Note: a matrix containing coordinates of a geometric figures is often called a coordinate matrix.

9. Yes, but are they equal? • Two matrices are considered equal if they have the same dimensions and each element of one matrix is equal to the corresponding element of the other matrix. This definition can be used to find values when elements of the matrices are algebraic expressions.

10. Definition of Equal Matrices • Two matrices are equal if and only if they have the same dimensions and their corresponding elements are equal.

11. Ex. 1: Solve for x and y Since the matrices are equal, the corresponding elements are equal. When you write the sentences that show this equality, two linear equations are formed. 6x = 62 + 8y Y = 6 – 2x

12. Ex. 1: Solve for x and y The second equation gives you a value for y that can be substituted into the first equation. Then you can find a value for x. 6x = 62 + 8y y = 6 – 2x 6x = 62 + 8y 6x = 62 + 8(6 – 2x) 6x = 62 + 48 – 16x 22x = 110 x = 5 Write down the first equation. Substitute 6 – 2x for y. Use the distributive property. Add 16x to both sides and combine like terms. Divide by 22.

13. Ex. 1: Solve for x and y To find a value for y, you can substitute 5 into either equation. y = 6 – 2x y = 6 – 2(5) y = 6 – 10 y = -4 Check your solutions by substituting values into the equation YOU DID NOT USE to find y. 6x = 62 + 8y 6(5) = 62 + 8(-4) 30 = 62 – 32 or 30✔ Write down the second equation. Substitute 5 for x. Use the distributive property. Simplify.

14. Scalar Multiplication • You can multiply any matrix by a constant. This is called scalar multiplication. When scalar multiplication is performed, each element is multiplied by that constant, and a new matrix is formed. This is summarized by the following rule:

15. Scalar Multiplication • Scalar multiplication of matrices can be used to find the coordinates of the vertices of a geometric figure that is enlarged or reduced. This type of change is called a dilation. When the size of a figure changes, the measures of its sides change in the same proportion. For example, if a figure triples in perimeter, its sides triple in length.

16. Graph ABC, then multiply the coordinates by 2. Ex. 2: Enlarge ABC, with vertices A(3, 2), B(-2, 1), C(1, -4), so that its perimeter is twice the perimeter of the original figure. The coordinates of the vertices of A’B’C’ are (6, 4), (-4, 2), (2, -8). The two triangles are similar. The perimeter of A’B’C’ is twice the perimeter of ABC

17. Addition of Matrices • Matrices can also be added. In order to add tow matrices, they must have the same dimensions. • If A and B are two m x n matrices, then A + B is an m x n matrix where each element is the sum of the corresponding elements of A and B.

18. Addition of Matrices • When a figure is moved from one location to another on the coordinate plane without changing its orientation, size, or shape, a translation occurs. You can use matrix addition to find the coordinates of the translated figures.

19. Ex. 3: Find the coordinates of quadrilateral QUAD if the figure is moved 5 units to the right and 1 unit down. 1. Write the coordinates of quadrilateral QUAD in the form of a matrix. 2. To translate the quadrilateral 5 units to the right means that each x-coordinate increases by 5. Translating the figure 1 unit down decreases each y-coordinate by 1.

20. Ex. 3: Find the coordinates of quadrilateral QUAD if the figure is moved 5 units to the right and 1 unit down. The matrix, called a translation matrix, that increases each x-value by five and decreases each y-value by 1 is: 3. To find the coordinates of the translated Q’U’A’D’, add the two matrices.

21. Ex. 3 continued: 4. Now graph the coordinates of quadrilateral Q’U’A’D’ to check the accuracy of your coordinates. The two quadrilaterals have the same size and shape. Quadrilateral Q’U’A’D’ is QUAD moved right 5 units and down 1 unit. The two figures are congruent.