1 C ollege A lgebra Systems and Matrices (Chapter5) 1

Cramer’s Rule for Two Equations in Two Variables

Example Use Cramer’s Rule to solve the system. 7x + 3y = 15 2x + 9y =  12 Find D first, since if D = 0, Cramer’s Rule does not apply. The solution set is {(3,  2)}.

General Form of Cramer’s Rule

Example Use Cramer’s Rule to solve the system.

Example continued Thus: The solution set is

Example

Solution: x = 2, y = -1, z = 3

5.4 Properties of Matrices

Basic Definitions It is customary to use capital letters to name matrices. Also, subscript notation is often used to name elements of a matrix, as shown. A n  n matrix is a square matrix. A matrix with just one row is a row matrix. A matrix with just one column is a column matrix. Two matrices are equal if they are the same size and if corresponding elements, position by position, are equal.

Example Find the values of the variables which makes the statement true. From the definition of equality, the only way that the statement can be true is if a = 3, b = 4, x = 2 and y = 7.

Addition and Subtraction of Matrices To add two matrices of the same size, add corresponding elements. Only matrices of the same size can be added. If A and B are two matrices of the same size, then A  B = A + (  B).

Examples Add and subtract the following. AddSubtract

Examples continued Add or subtract, if possible. a) b) The matrices have different sizes so they cannot be added or subtracted.

Properties of Scalar Multiplication If A and B are matrices of the same size and c and d are scalars, then (c + d)A = cA + dAc(A)d = cd(A) c(A + B) = cA + cB(cd)A = c(dA) Example: Find the product.

Matrix Multiplication If the number of columns of an m  n matrix A is the same as the number of rows of an n  p matrix B (i.e., both n). The element c ij of the product matrix C = AB is found as follows: Matrix AB will be an m  p matrix.

Example Suppose A is a 3  4 matrix, while B is a 4  2 matrix. a) Can the product AB be calculated? b) If AB can be calculated, what size is it? c) Can BA be calculated? d) If BA can be calculated, what size is it?

Solutions a) AB can be calculated, because the number of columns of A is equal to the number of rows of B. 3  4 4  2 b) The product is a 3  2 matrix. c) BA cannot be calculated, the number of columns and rows do not match. d) Since BA cannot be calculated we cannot determine the size. must match Size of AB

Multiply the Matrices For find each of the following. a) AB b) BA c) AC

Solution AB A is a 2  3 matrix and B is a 3  2 matrix, so AB will be a 2  2 matrix.

Solution BA B is a 3  2 matrix and A is a 2  3 matrix, so BA will be a 3  3 matrix.

Solution AC The product AC is not defined because the number of columns of A, 3, is not equal to the number of rows of C, 2. Note that AB  BA. Multiplication of matrices is generally not commutative.

Properties of Matrix Multiplication If A, B, and C are matrices such that all of the following products and sums exist, then (AB)C = A(BC) A(B + C) = AB + AC (B + C)A = BA + CA.

A matrix with m rows and n columns is called an m by n matrix.

Note that in order for two matrices to be combined with addition or subtraction, they must have the same number of rows and columns.

If k is a real number and A is an m by n matrix, the matrix kA is

Let A denote an m by r matrix and let B denote an r by n matrix. The product AB is defined as the m by n matrix whose entry in row i, column j is the product of the ith row of A and the jth column of B.

5.6 Matrix Inverses

Identity Matrices By the identity property for real numbers, a  1 = a and 1  a = a for any real number a. If there is to be a multiplicative identity matrix I, such that AI = A and IA = A for any matrix A, then A and I must be square matrices of the same size.

Identity Matrices 2  2 Identity Matrix If I 2 represents the 2  2 identity matrix, then

Identity Matrices

Stating and Verifying the 3  3 Identity Matrix Example: Let Give the 3  3 identity matrix I and show that AI = A. Solution: By the definition of matrix multiplication,

Multiplicative Inverses If A is an n  n matrix, then its multiplicative inverse, written A  1, must satisfy both AA  1 = I n and A  1 A = I n This means that only a square matrix can have a multiplicative inverse. Caution: Although for any nonzero real number a, if A is matrix,.

Finding an Inverse Matrix To obtain A  1 for any n  n matrix A for which A  1 exists, follow these steps. Step 1 Form the augmented matrix where I n is the n  n identity matrix.. Step 2 Perform row transformations on to obtain a matrix of the Step 3 Matrix B is A  1.

Example Find A  1 if A = Solution: Use row transformations as follows. Step 1 Write the augmented matrix Step 2 Since 1 is already in the upper left hand corner, we begin by using row transformation that will result in 0 for the first element in the second row.

The following is done to obtain 0 as the first element in the third row. The following is done to obtain 1 as the third element in the third row. Example continued R2  2R1 R3 + R1 The following is done to obtain 0 for the third element in the first row. R3  R3 R1  R3

Example continued The following is done to obtain 0 for the third element in the second row. Step 3 The last transformation shows that the inverse is R2  R3

Solution of the Matrix Equation AX = B If A is an n  n matrix with inverse A  1, X is an n  1 matrix of variables, and B is an n  1 matrix, then the matrix equation AX = B has the solution GX = A  1 B.

Solving Systems of Equations Using Matrix Inverses Example: 3x + 4y = 5 5x + 7y = 9 Solution: To represent the system as a matrix equation, use one matrix for the coefficients, one for the variables, and one for the constants, as follows The system can then be written in matrix form as the equation AX = B, since

Solving Systems of Equations Using Matrix Inverses continued To solve the system, first find A  1. Then find A  1 B. Since X = A  1 B, The final matrix shows the solution set of the system is {(  1, 2)}

Continue performing row operations on the augmented matrix until the matrix on the left is the identity matrix.

If a matrix represents the coefficients of a linear system of equations, the inverse matrix can be used to solve the system.

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