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Matrix Inverses

Augmented Matrix:

Row operations:

0 1 0
1 0 0
0 0 1
Interchange rows       and   

2 0 0
0 1 0
0 0 1
Multiply row       by    /

1 0 0
2 1 0
0 0 1
Add    /    times row       to row   

Original Matrix:

Row Operations Performed:

About Matrix Inverses

The inverse of an \(n \times n\) matrix \(M\) is another \(n \times n\) matrix, denoted \(M^{-1}\) such that \[ MM^{-1} = M^{-1}M = I \] where \(I\) is the \(n \times n\) identity matrix. Suppose a square matrix can be reduced to the identity by a sequence of elementary row operations. The inverse of this matrix is the result of applying the same operations, in the same order, to the identity.

Some important facts regarding matrix inverses include:

Using the Applet

This applet shows one method of finding the inverse of a randomly generated \(3 \times 3\) invertible matrix. Use the controls below the augmented matrix to perform elementary row operations. The process is complete when the left side is reduced to the identity.

On large screens, the elementary matrix of the operation will be displayed, and the operations performed will be recorded next to the augmented matrix.

About this Applet

This applet was created using JavaScript and the Raphael library. If you are unable to see the applet, make sure you have JavaScript enabled in your browser. This applet may not be supported by older browsers.