Matrix Solutions to Linear Systems A matrix is a rectangular array of elements. o An array is a systematic arrangement of numbers or symbols in rows and columns. Matrices (the plural of matrix) may be used to display information and to solve systems of linear equations. The numbers in the rows and columns of a matrix are called the elements of the matrix. Matrices are rectangular arrays of numbers that can aid us by eliminating the need to write the variables at each step of the reduction. For example, the system 2 x + 4 y + 6 z 22 3 x + 8 y + 5 z 27 x + y + 2 z 2 may be represented by the augmented matrix Dimensions of a Matrix The dimensions of a matrix may be indicated with the notation r s, where r is the number of rows and s is the number of columns of a matrix. A matrix that contains the same number of rows and columns is called a square matrix. Example: 3 3 square matrices: 1 Page
Example - Write the augmented matrix of the given system of equations. 3x+4y7 4x-2y5 Example - Write the system of equations corresponding to this augmented matrix. Then perform the row operation on the given augmented matrix. R 4r + r 1 3 3 5 4 5 3 5 3 2 4 6 2 1 2 Augmented Matrix The first step in solving a system of equations using matrices is to represent the system of equations with an augmented matrix. o An augmented matrix consists of two smaller matrices, one for the coefficients of the variables and one for the constant Systems of equations a 1 x + b 1 y c 1 a 2 x + b 2 y c 2 Augmented Matrix a b c a b c 1 1 1 2 2 2 Row Transformations To solve a system of equations by using matrices, we use row transformations to obtain new matrices that have the same solution as the original system. We use row transformations to obtain an augmented matrix whose numbers to the left of the vertical bar are the same as the multiplicative identity matrix. 2 Page
Procedures for Row Transformations Any two rows of a matrix may be interchanged. All the numbers in any row may be multiplied by any nonzero real number. All the numbers in any row may be multiplied by any nonzero real number, and these products may be added to the corresponding numbers in any other row of numbers. A matrix with 1 s down the main diagonal and 0 s below the 1 s is said to be in row echelon form (ref). We use row operations on the augmented matrix. These row operations are just like what you did when solving system of equations using addition method, this method is called Gaussian Elimination. A matrix with 1 s down the main diagonal and 0 s above and below the 1 s is said to be in reduced row echelon form (rref). We use row operations on the augmented matrix. These row operations are just like what you did when solving system of equations using addition method. To Change an Augmented Matrix to the Reduced Row Echelon Form (rref) Use row transformations to: 1. Change the element in the first column, first row to a 1. 2. Change the element in the first column, second row to a 0. 3. Change the element in the second column, second row to a 1. 4. Change the element in the second column, first row to a 0. Sometimes it is advantageous to write a matrix in reduced row echelon form. In this form, row operations are used to obtain entries that are 0 above as below the leasing 1 in a row. The advantage is that the solution is readily found without needing to back substitute. 3 Page
Graphing Calculator-Matrices, Reduced Row Echelon Form To work with matrices we need to press 2 nd x -1 key to get to the matrix menu. To type in a matrix, cursor to the right twice to get to EDIT, press ENTER and type in the dimensions of matrix A. Press ENTER after each number, then type in the numbers that comprise the matrix, again pressing ENTER after each number. To get out of the matrix menu press QUIT (2 nd MODE) To get to reduced row echelon form bring up the matrix menu (2 nd x -1) and cursor to MATH. Cursor down to B, rref (reduced row echelon form). Press ENTER To type the name of the matrix you want to work with, again bring up the matrix menu (2 nd x - 1) and under names, choose the letter of the matrix you are working with, press ENTER again. 4 Page
Example: Solve this system of equations using matrices (row operations). 3x 5y 3 15x+ 5y 21 Example: Solve the system of equations using matrices. If the system has no solution, say inconsistent. Example: Write the system of equations corresponding to this augmented matrix. Then perform the row operation on the given augmented matrix. R 4r + r 1 3 3 5 4 5 3 5 3 2 4 6 2 1 2 5 Page
A System of Equations with an Infinite Number of Solutions Example: Solve the system of equations given by x + 2 y 3 z 2 3 x y 2 z 1 2 x + 3 y 5 z 3 A System of Equations That Has No Solution Example: Solve the system of equations given by x + y + z 1 3 x y z 4 x + 5 y + 5 z 1 Systems with no Solution If there is a row in the augmented matrix containing all zeros to the left of the vertical line and a nonzero entry to the right of the line, then the system of equations has no solution. 6 Page
Determinants and Cramer s Rule The Determinant of a 2 x 2 Matrix Example : Evaluate the determinant of each of the following matices: 2 3 a. 5 1 3 2 b. 4 1 7 Page
To evaluate determinates in your calculator. Press 2 nd x -1 then scroll over to EDIT to input your matrix. Then 2nd MODE to get out of the matrix menu. Then go back into the matrix menu 2 nd x -1 scroll over to MATH and down to det (determinate). Press ENTER and then go back into matrix menu to call the matrix that you want ot take the determinate of. Example : Evaluate the determinant of the following matrix: 2 1 0 1 1 2 3 1 0 8 Page
The Determinant of Any N x N Matrix The determinant of a matrix with n rows and n columns is said to be an nth-order determinant. The value of an nth-order determinant can be found in terms of determinants of order n-1. We can generalize the idea for fourth-order determinants and higher. We have seen that the minor of the element a is the determinant obtained by deleting the ith row and the jth column in the given array of numbers. The cofactor of the element a i+ j is (-1) times the minor of the a entry. If the sum of the row and column (i+j) is even, the cofactor is the same as the minor. If the sum of the row and column (i+j) is odd, the cofactor is the opposite of the minor. ij th ij ij 9 Page
Example: Find the determinate of the following matrix. 10 Page