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Chapter 3: Q20P (page 123)

solve the set of equations by the method of finding the inverse of the coefficient matrix.

{2x+3y=-15x+4y=8

Short Answer

Expert verified

The solution of the given set of equations is x=4 and y=-3 .

Step by step solution

01

Definition of inverse of a matrix:

The inverse of a matrix M is defined as the matrix M-1 such that MM-1 and M-1M are both equal to a unit matrix I .

02

 Find the inverse of the matrix:

The given set of equations are

2x+3y=-15x+4y=8

The solution is to be found by the method of finding the inverse of the coefficient matrix. Here,

M=2354

So,

C=C11C12C21C22=4-5-32

Find inverse of matrix M .

M-1=1detMCT=14脳2-3脳54-3-52=17-435-2

03

 Find the solution of equations:

Use equation M-1k=rto solve the equations, where is the matrix of coefficient and is the unknown vector. Here, k=-18 and r=xy .

17-435-2-18=xy1728-21=xy4-3=xy

Therefore, x=4 and y=-3 .

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Most popular questions from this chapter

Find AB,BA,A+B,A-B,A2,B2,5-A,3-B. Observe thatAB鈮�BA.Show that(A-B)(A+B)鈮�(A+B)(A-B)鈮�A2-B2. Show that det(AB)=det(BA)=(detA)(detB), but that det(A+B)鈮�detA+detB. Show that det(5A)鈮�5detA and find n so that localid="1658983435079" det(5A)=5ndetA.Find similar results for det(3B). Remember that the point of doing these simple problems by hand is to learn how to manipulate determinants and matrices correctly. Check your answers by computer.

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Find the Eigen values and eigenvectors of the following matrices. Do some problems by hand to be sure you understand what the process means. Then check your results by computer.

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A particle is traveling along the line (x-3)/2=(y+1)/(-2)=z-1. Write the equation of its path in the form r=r0+At. Find the distance of closest approach of the particle to the origin (that is, the distance from the origin to the line). If t represents time, show that the time of closest approach is t=-(r0脳A)/|A|2. Use this value to check your answer for the distance of closest approach. Hint: See Figure 5.3. If P is the point of closest approach, what is A脳r2?

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Question: For each of the following problems write and row reduce the augmented matrix to find out whether the given set of equations has exactly one solution, no solutions, or an infinite set of solutions. Check your results by computer. Warning hint:Be sure your equations are written in standard form. Comment: Remember that the point of doing these problems is not just to get an answer (which your computer will give you), but to become familiar with the terminology, ideas, and notation we are using

6.{x+y-z=13x+2y-2z=3
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