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Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

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Chapter
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BuyFindarrow_forward

Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

SKILLS

3 9 4 6 . Solving a linear system as a matrix equation Solve the system of equations by converting to a matrix equation and using the inverse of the co-efficient matrix, as in example 6. Use the inverses from Exercises 11 14 , 19 , 20 , 23 , and 25 .

{ x + 2 y + 3 w = 0 y + z + w = 1 y + w = 2 x + 2 y + 2 w = 3

1 1 2 6 . Finding the inverse of a matrix Find the inverse of a matrix if it exists.

2 5 . [ 1 2 0 3 0 1 1 1 0 1 0 1 1 2 0 2 ]

To determine

To solve:

The system of equations {x+2y+3w=0y+z+w=1y+w=2x+2y+2w=3 by converting it into a matrix equation and find the values of x, y, z, and w.

Explanation

Approach:

A system of linear equations can be written in the form of AX=B where A is called the coefficient matrix, B is called the known matrix, and X is called the variable matrix.

If a square matrix A of dimension 3×3 has an inverse A1, and if X is a variable matrix and B is a known matrix both with n rows, then the solution of the matrix equation AX=B is given by:

X=A1B.

Take a 4×8 matrix with left half as matrix B and right half as I4 and transform the left half into identity matrix by performing elementary row operations on the entire matrix, the right half becomes inverse of the matrix B.

Calculation:

The system of linear equations {x+2y+3w=0y+z+w=1y+w=2x+2y+2w=3 can be written as a single matrix equation AX=B as follows.

[1203011101011202][xyzw]=[0123]

Here A=[1203011101011202], B=[0123], and X=[xyzw].

The solution of this matrix equation is given by X=A1B.

First find A1,

Take a matrix of dimension 4×8 with left half as matrix A and right half as I4 and perform row operations to transform the left half into I4.

[ 12031000011101000101001012020001 ]R4R4R1[ 120310000111010001010010000 1 1001 ]R3R3R2[ 120310000111010000 100 110000 1 1001 ]R1R12R2[ 10 211 2000111010000 100 110000 1 1001 ]R3R3[ 10 211 20001110100001001 10000 1 1001 ]

R2R2R3[

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