Linear System of Simultaneous Equations Warm UP First precinct: 6 arrests last week equally divided between felonies and misdemeanors. Second precinct: 9 arrests - there were twice as many felonies as the first precinct. Write a system of two equations and find out how many felonies and misdemeanors occurred.

Sections 4.1 & 4.2 Matrix Properties and Operations

Algebra

Matrix A matrix is any doubly subscripted array of elements arranged in rows and columns enclosed by brackets.

Dimensions of a Matrix

Name the Dimensions

Row Matrix [1 x n] matrix

Column Matrix [m x 1] matrix

Square Matrix Same number of rows and columns Matrices of nth order -B is a 3 rd order matrix

The Identity

Identity Matrix Square matrix with ones on the diagonal and zeros elsewhere.

The Equal

Equal Matrices Two matrices are considered equal if they have the same dimensions and if each element of one matrix is equal to the corresponding element of the other matrix. = Can be used to find values when elements of an equal matrices are algebraic expressions

To solve an equation with matrices 1. Write equations from matrix 2. Solve system of equations Examples =

Linear System of Simultaneous Equations How can we convert this to a matrix?

Matrix Addition A new matrix C may be defined as the additive combination of matrices A and B where: C = A + B is defined by: Note: all three matrices are of the same dimension

Addition IfIf and then

Matrix Subtraction C = A - B Is defined by

Subtraction IfIf and then

Matrix Addition Example

Multiplying Matrices by Scalars

Matrix Operations

Matrix Multiplication Matrices A and B have these dimensions: Video [r x c] and [s x d]

Matrix Multiplication Matrices A and B can be multiplied if: [r x c] and [s x d] c = s

Matrix Multiplication The resulting matrix will have the dimensions: [r x c] and [s x d] r x d

A x B = C [2 x 2] [2 x 3]

A x B = C [3 x 2][2 x 3] A and B can be multiplied [3 x 3] [3 x 2][2 x 3] Result is 3 x 3

Practice

Combing Steps

Matrices In the Calculator 2 nd x -1 button must enter dimensions before data must enter the matrix before doing calculations

2.5 Determinants of 2 X 2 Matrix

Example Find the value of = 3(9) - 2(5) or 17

Determinants of 3 X 3 Matrix Third-order determinants - Determinants of 3 × 3 matrices are called Expansion by diagonals Step 1: begin by writing the first two columns on the right side of the determinant, as shown below Step 2: draw diagonals from each element of the top row of the determinant Downward to the right. Find the product of the elements on each diagonal.

Determinants of 3 X 3 Matrix Step 4: To find the value of the determinant, add the products of the first set of diagonals and then subtract the products of the second set of diagonals. The value is: Step 3: draw diagonals from the elements in the third row of the determinant upward to the right. Find the product of the elements on each diagonal.

Example

Determinants of 3 X 3 Matrix expansion by minors. The minor of an element is the determinant formed when the row and column containing that element are deleted.

Example. = 2 -(-3) +(-5) = 2(-8) + 3(-11) – 5(-7) = -14

Inversion

Matrix Inversion Like a reciprocal in scalar math Like the number one in scalar math

Inverses

2.5 Inverses Step 1 : Find the determinant. Step 2 : Swap the elements of the leading diagonal. Recall: The leading diagonal is from top left to bottom right of the matrix. Step 3: Change the signs of the elements of the other diagonal. Step 4: Divide each element by the determinant.

Example First find the determinant = 4(2) - 3(2) or 2 or..

Solving Systems with Matrices Step 1: Write system as matrices Step 2: Find inverse of the coefficient matrix. Step 3: Multiply each side of the matrix equation by the inverse Coefficient Matrix Variable Matrix Constant Matrix

Example Solve the system of equations by using matrix equations. 3x + 2y = 3 2x – 4y = 2

Example Write the equations in the form ax + by = c 2x – 2y – 3 = 0 ⇒ 2x – 2y = 3 8y = 7x + 2 ⇒ 7x – 8y = –2 Step 2: Write the equations in matrix form. Step 3: Find the inverse of the 2 × 2 matrix. Determinant = (2 × –8) – (–2 × 7) = – 2 Step 4: Multiply both sides of the matrix equations with the inverse

Using the calculator How do you use the calculator to find the solution to a system of equations? Put both coefficient and answer matrix into calculator Multiply the inverse of the coefficient matrix and the answer matrix to get values. 3x + 2y = 3x + 2y+3z=5 2x – 4y = 23x+2y-2z=-13 5x+3y-z=-11

Modeling Motion with Matrices Vertex Matrix – A matrix used to represent the coordinates of the vertices of a polygon Transformations -Functions that map points of a pre-image onto its image Preimage-image before any changes Image-image after changes Isometry-a transformation in which the image and preimage are congruent figures Translation – a figure moved from one location to another without cahnging sizze, shape, or orientation

Translation Suppose triangle ABC with vertices A(-3, 1), B(1, 4), and C(-1, -2) is translated 2 units right and 3 units down. a.Represent the vertices of the triangle as a matrix. b.Write the translation matrix. c.Use the translation matrix to find the vertices of A’B’C’, the translated image of the triangle. d.Graph triangle ABC and its image.

Translation a.The matrix representing the coordinates of the vertices of triangle ABC will be a 2  3 matrix. b.The translation matrix is. c.Add the two matrices. d.Graph the points represented by the resulting matrix.

Example ΔX'Y'Z' is the result of a translation of ΔXYZ. A table of the translations is shown. Find the coordinates of Y and Z'. Solve the Test Item Write a matrix equation. Let (a, b) represent the coordinates of Y and let (c, d) represent the coordinates of Z'. Since these two matrices are equal, corresponding elements are equal. Solve an equation for x. Solve an equation for y. –3 + x = y = 7 x = 7 y = 5 Use the values for x and y to find the values for Y(a, b) and Z' (c, d). a = –4 b = –5 9 = c 4 = d ΔXYZΔX'Y'Z' X(–3, 2)X'(4, 7) YY'(3, 0) Z(2, –1)Z'Z'

Dilation When a geometric figure is enlarged or reduced ALL linear measures of the image change in the same ration

Example Quadrilateral DEFG has vertices D(1, 2), E(4, 1), F(3, –1), and G(0, 0). Dilate quadrilateral DEFG so that its perimeter is two and one–half times the original perimeter. What are the coordinates of the vertices of quadrilateral D'E'F'G'? If the perimeter of a figure is two and one–half times the original perimeter, then the lengths of the sides of the figure will be two and one–half times the measure of the original lengths.  Multiply the vertex matrix by the scalar 2.5. The coordinates of the vertices of quadrilateral D'E'F'G' are D'(2.5, 5), and E'(10, 2.5), F'(7.5, –2.5), and G'(0, 0).

2.4 Modeling Motion with Matrices Reflection Matrices X-Axis Y-Axis Line y=x

Reflection Example Find the coordinates of the vertices of the image of quadrilateral ABCD with A(–2, 1), B(–1, 4), C(3, 2), and D(4, –2) after a reflection across the line y = x. Write the ordered pairs as a vertex matrix. Then multiply the vertex matrix by the reflection matrix for the line y = x. The coordinates of the vertices of A'B'C'D' are A'(1, –2), B'(4, – 1), C'(2, 3), and D'(–2, 4). Notice that the preimage and image are congruent. Both figures have the same size and shape.

2.4 Modeling Motion with Matrices Rotation Matrices 90 degrees 180 degrees 270 degrees

Rotation Example Find the coordinates of the vertices of the image of quadrilateral MNOP with M(2, 2), N(2, 5), O(3, 4), and P(4, 1) after it is rotated 270° counterclockwise about the origin. Write the ordered pairs in a vertex matrix. Then multiply the vertex matrix by the rotation matrix. The coordinates of the vertices of quadrilateral M'N'O'P' are M'(2, – 2), N'(5, –2), O'(4, –3), and P'(1, – 4). The image is congruent to the preimage