# Solving a system with the inverse of a matrix.

• thatguythere
In summary: So now I am completely lost.In summary, the conversation discusses a problem involving using Gauss-Jordan elimination to find the inverse of a given matrix and using the result to solve a system of equations. However, there is confusion about how the inverse can be used to solve the system, and it is later determined that there was a mistake in the problem.
thatguythere

## Homework Statement

a)Use Gauss-Jordan elimination to find the inverse of A =
[ 2 1 4 ]
[ 1 1 2 ]
[ -2 -3 -2 ]

b) Use the result from part a) to find the solution of the following system.

5x+2y-3z = 5
x+y-z = -1
-3x-y+2z = 2

## The Attempt at a Solution

My problem is not with part a), I quite easily found the inverse of the matrix. What I am not understanding is what exactly they are asking me to do for part b). How can I use the inverse of one matrix to solve for another? Any help is greatly appreciated.

If you change

5x+2y-3z = 5
x+y-z = -1
-3x-y+2z = 2

into a matrix equation in the form of Bx=C, what would B, X and C be?

rock.freak667 said:
If you change

5x+2y-3z = 5
x+y-z = -1
-3x-y+2z = 2

into a matrix equation in the form of Bx=C, what would B, X and C be?

I'm with thatguythere. I can't see that the matrix in part b) is related in any simple way to the matrix in part a). I'm suspecting the somebody goofed when assembling the problem.

I contacted my TA and it is indeed a mistake. It should be
2x + y + 4z = 5
x + y + 2z = -1
-2x -3y -2z = 2
I should be able to manage now, thanks.

Dick said:
I'm with thatguythere. I can't see that the matrix in part b) is related in any simple way to the matrix in part a). I'm suspecting the somebody goofed when assembling the problem.

Ah well, I didn't calculate the inverse, so I assumed it one of those problems where the matrix equation would be A-1x=B

## 1. What is the inverse of a matrix?

The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. In other words, it "undoes" the original matrix and is denoted as A-1 for a matrix A.

## 2. Why is the inverse of a matrix important?

The inverse of a matrix is important because it allows us to "solve" a system of linear equations. Instead of using traditional methods like elimination or substitution, we can use the inverse of the coefficient matrix to find the solution to the system.

## 3. How do you find the inverse of a matrix?

The inverse of a matrix can be found by using various methods such as Gauss-Jordan elimination, Cramer's rule, or the adjugate method. These methods involve performing operations on the original matrix to transform it into its inverse.

## 4. When can a matrix not have an inverse?

A matrix does not have an inverse if it is singular, meaning its determinant is 0. This can happen if the matrix is not square or if its columns are linearly dependent. In these cases, the system of equations has either no solution or infinitely many solutions.

## 5. Can the inverse of a matrix be used to solve any system of equations?

No, the inverse of a matrix can only be used to solve systems of linear equations. It is not applicable for nonlinear systems of equations. Additionally, the inverse method may not always be the most efficient or accurate method for solving a system, so it is important to consider other options as well.

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