Question 1: How do you write a system of equations as a matrix equation?
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1 Question : How do ou write a sstem of equations as a matrix equation? In section 3.2, ou learned how to multipl two matrices. This process involved multipling the entries in the row of one matrix b the columns of another matrix and then adding those products. For instance, suppose we multipl 25 6 Since we are multipling a 2 x 2 matrix b a 2 x matrix, the product is a 2 x matrix. Working out the product, we get When the product is written this wa, we can easil identif the product of the numbers in the rows of the first matrix with the numbers in the second matrix as well as the sum. Instead of multipling the first matrix b a constant column matrix, let s tr multipling b a matrix containing two variables x and, x When we carr out the multiplication now, we get 2x x 2 3x 4 The entries in the product look a lot like the part of a sstem of equations. In fact, if we set this equal to a 2 x matrix with constants, 2x 7 39 we can write the matrix equation as an equivalent sstem of equations. These matrices are equal when the corresponding entries are equal or when 2
2 x 2 7 3x 4 39 In other words, 2x 7 x 2 7 is the same as 39 3x 4 39 When a sstem of equations is written in terms of matrices, we call it a matrix equation. In this matrix equation, the three matrices are tpicall called A, X and B. 2x 7 39 A X B The matrix A is called the coefficient matrix and it entries are the coefficients on the variables when the are written in the same order in each equation. The matrix X is called the variable matrix and contains the two variables in the problem. The matrix B is called the constant matrix and contains the constants from the right hand side of the matrix equation. To match the different matrices with their entries, each equation must have the variables on the left side with the variable terms listed in the same order. The constants must be on the right side. Once each equation has this format, we can read the entries in each matrix. Example Write a Linear Sstem as a Matrix Equation Write the sstem of linear equations x 5 as the matrix equation AX B. 3x4 3
3 Solution The sstem is in the proper format to determine the coefficient matrix and the constant matrix. The variable matrix for this sstem is X x Using the coefficients on the variables, define the coefficient matrix and constant matrix as 5 A B With these definitions, the product AX is equal to AX x x 3x 4 When this matrix is set equal to the constant matrix B, we get x 5 3x 4 These matrices are equal when x 5 and 3x 4. This means the matrix equation AX equations. B is equivalent to the original sstem of We can use the same strateg to write larger sstems of equations as matrix equations. This leads to larger matrices. We incorporate different variable names into the variable matrix X as shown in the next example. 4
4 Example 2 Write a Linear Sstem as a Matrix Equation Write the sstem of linear equations x x x 2 2x 2x 3x 2x 7 x 3 as a matrix equation AX B. Solution Before we can define the matrices, we need to put each equation in the proper format. In the first and second equations, all variable terms are on one side of the equation and the constant terms are on the other side of the equations. In the third equation, there are variable terms on both sides of the equation. We can put the third equation in the proper format b subtracting x from both sides of the equation, 2x x 7x x 3 x 2x 7 3 This leads to the sstem x x x 2 2x 2x 3x x 2x 7 3 The variable matrix is X x x 2 x 3 The coefficients on the variables give the coefficient matrix, 5
5 A Note that an signs are included in the coefficients and variables that are missing correspond to a coefficient of zero. The constant matrix, 2 B 7 matches the constants on the right hand of the sstem. We can check to see that the matrix equation AX the sstem b carring out the product AX : B is equivalent to x xx2 x3 AX x 2x 2x 3x x 3 x2x 3 If we set this matrix equal to B, xx2 x3 2 2x 2x 3x x2x 3 7 the corresponding entries must be equal. This is equivalent to the original sstem after it had been modified to put it into the proper format. 6
6 Once a sstem of linear equations is written as a matrix equation AX B, we can use the inverse of the matrix A to find the solution to the sstem. In the next question, we ll learn how to do this and use it to solve an application. 7
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