Unit+4

=Unit 4 Matrices=

Vocabulary

 * augmented matrix (p. 218) || matrix (p. 164) || rotation (p. 190) ||
 * center of rotation (p. 190) || matrix addition (p. 170) || row operations (p. 219) ||
 * coefficient matrix (p. 210) || matrix element (p. 165) || scalar (p. 178) ||
 * constant matrix (p. 210) || matrix equation (p. 172) || scalar product (p. 178) ||
 * Cramer's Rule (p. 217) || matrix multiplication (p. 180) || square matrix (p. 195) ||
 * determinant (p. 196) || multiplicative identity matrix (p. 195) || transformation (p. 188) ||
 * dilation (p. 188) || multiplicative inverse matrix (p. 195) || translation (p. 188) ||
 * equal matrices (p. 173) || preimage (p. 188) || variable matrix (p. 210) ||
 * image (p. 188) || reflection (p. 189) || zero matrix (p. 171) ||

4-1 Objectives
It is often useful to organize data into matrices. A **matrix** is a rectangular array of numbers classified by its dimensions. An //m// × //n// matrix has //m// rows and //n// columns. A **matrix element** //a////ij// is in the //i//th row and //j//th column of matrix //A//.
 * 1) To identify and classify matrices and their elements (p. 164)
 * 2) To organize data into matrices (p. 165)

4-2 and 4-3 Objectives
To perform **matrix addition** or subtraction, add or subtract the corresponding elements in the matrices. To obtain the **scalar product** of a matrix and a **scalar,** multiply each matrix element by the scalar. **Matrix multiplication** uses both multiplication and addition. The element in the //i//th row and //j//th column of the product of two matrices is the sum of the products of each element of the //i//th row of the first matrix and each element of the //j//th column of the second matrix. The first matrix must have the same number of columns as the second has rows. Two matrices are **equal matrices** when corresponding elements are equal and they have the same dimensions. This principle is used to solve a **matrix equation.**
 * 1) To add and subtract matrices (p. 170)
 * 2) To solve some matrix equations (p. 172)
 * 3) To multiply a matrix by a scalar (p. 178)
 * 4) To multiply two matrices (p. 180)

4-4 Objectives
A **transformation** is a change made to a figure. The original figure is the **preimage,** and the transformed figure is the **image.** A **translation** slides a figure without changing its size or shape. A **dilation** changes the size of a figure. You can use matrix addition to translate a figure and scalar multiplication to dilate a figure. You can use multiplication by the appropriate matrix to perform transformations that are specific **reflections** or **rotations.** For example, to reflect a figure in the //y//-axis, multiply by. To rotate a figure 180 °, multiply by.
 * 1) To represent translations and dilations with matrices (p. 187)
 * 2) To represent reflections and rotations with matrices (p. 189)



4-5, 4-6 and 4-7 Objectives
A **square matrix** with 1's along its main diagonal and 0's elsewhere is the **multiplicative identity matrix,** //I//. If //A// and //X// are square matrices such that //AX// = //I//, then //X// is the multiplicative inverse matrix of //A//, //A// − 1. You can use formulas to evaluate the determinants of 2 × 2 and 3 × 3 matrices. You can use a calculator to find the inverse of a matrix. The inverse of a 2 × 2 matrix can be found by using its determinant. You can use inverse matrices to solve some matrix equations. You can also use inverse matrices to solve some systems of equations. When equations in a system are in standard form, the product of the **coefficient matrix** and the **variable matrix** equals the **constant matrix.** You solve the equation by multiplying both sides of the equation by the inverse of the coefficient matrix. If that inverse does not exist, the system does not have a unique solution.
 * 1) To evaluate determinants of 2 × 2 matrices and find inverse matrices (p. 195)
 * 2) To use inverse matrices in solving matrix equations (pp. 197 and 203)
 * 3) To evaluate determinants of 3 × 3 matrices (p. 202)
 * 4) To solve systems of equations using inverse matrices (p. 210)

4-8 Objectives
You can also use **row operations** on an augmented matrix to solve a system.
 * 1) To solve a system of equations using Cramer's Rule (p. 217)
 * 2) To solve a system of equations using augmented matrices (p. 218)
 * Cramer's Rule** for solving systems of equations uses determinants to solve for each variable. //D// is the determinant of the coefficient matrix. //D////y// is the determinant formed by replacing the coefficients of //y// in //D// with the constant terms.