To determine To calculate: Use the inverse of the coefficient matrix to solve each system of equation. Explanation Given: The system of equations: 2 x + 2 y − 4 z = 12 2 x + 6 y = 16 − 3 x − 3 y + 5 z = − 20 Formula used A system of equations with the same number of equations as variables is written in matrix form as AX = B, where A is the square matrix of coefficients, X is the column matrix of variables, and B is the column matrix of constants. If A has an inverse, then the unique solution of the system is X = A -1 B. Calculation: Consider the provided system of equations: 2 x + 2 y − 4 z = 12 2 x + 6 y = 16 − 3 x − 3 y + 5 z = − 20 Now create a 3 × 3 matrix with the coefficients of the variable provided in the system of equation. Therefore, A = [ 2 2 − 4 2 6 0 − 3 − 3 5 ] Similarly, form a matrix 3 × 1 of the constants written on the right-hand side of the provided system of equations. Therefore, B = [ 12 16 − 20 ] Now rewrite the equation as, A X = B Where X = [ x y z ] So, the solution is X = A − 1 B . Now to solve this matrix equation, calculate the inverse of the matrix A . As, A A − 1 = Ι So, [ 2 2 − 4 2 6 0 − 3 − 3 5 ] A − 1 = [ 1 0 0 0 1 0 0 0 1 ] Use the transformation R 3 → 2 R 3 + 3 R 1 , [ 2 2 − 4 2 6 0 − 6 + 6 − 6 + 6 10 − 12 ] A − 1 = [ 1 0 0 0 1 0 0 + 3 0 + 0 1 + 0 ] [ 2 2 − 4 2 6 0 0 0 − 2 ] A − 1 = [ 1 0 0 0 1 0 3 0 1 ] Use the transformation R 1 → R 1 − 2 R 3 , [ 2 − 0 2 − 0 − 4 + 4 2 6 0 0 0 − 2 ] A − 1 = [ 1 − 6 0 − 0 0 − 2 0 1 0 3 0 1 ] [ 2 2 0 2 6 0 0 0 − 2 ] A − 1 = [ − 5 0 − 2 0 1 0 3 0 1 ] Use the transformation R 2 → R 2 − R 1 , [ 2 2 0 2 − 2 6 − 2 0 − 0 0 0 − 2 ] A − 1 = [ − 5 0 − 2 0 + 5 1 − 0 0 + 2 3 0 1 ] [ 2 2 0 0 4 0 0 0 − 2 ] A − 1 = [ − 5 0 − 2 5 1 2 3 0 1 ] Use the transformation R 1 → 2 R 1 − R 2 [ 4 − 0 4 <

12th Edition

ISBN: 8220106844557

Author: MULLINS

Publisher: PEARSON

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Matrices

‘Matrices’ is the plural form of the word matrix, and it is basically a spreadsheet in the form of a box. In mathematics, various functions can be carried out with matrices. Generally, a matrix comes in the shape of a square or rectangle. The elements ar…

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Chapter 6.6, Problem 12E

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To calculate: Use the inverse of the coefficient matrix to solve each system of equation.

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Chapter 6.6, Problem 11E

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