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The-Inverse-of-a-Square-Matrix

This version was saved 15 years, 4 months ago View current version     Page history
Saved by Elizabeth Dauterman
on December 18, 2008 at 4:56:37 pm
 

The Inverse of a Square Matrix

 

Invertible/Nonsingular Matrix - any matrix that has an inverse

 

Singular Matrix - any matrix that does not have an inverse; either a non-square matrix or a square matrix with a determinant of 0 (he

 

An invertible matrix multiplied by its inverse will produce the identity matrix. This is a square matrix which is indicated by I. It has 1s along the diagonal from the top-left corner of the matrix to the bottom-right corner, along with 0s in all the other spots.

 

Identity matrices look like this: 

I_1=\begin{bmatrix} 1 \end{bmatrix} , I_2=\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} , I_3=\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} ,\ \cdots , I_n=\begin{bmatrix} 1 & 0 & \cdots & 0 \ 0 & 1 & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & 1 \end{bmatrix}

 

To show that two matrics are inverses, you should multiply them (http://www.easycalculation.com/matrix/learn-matrix-multiplication.php) and produce the identity matrix.

 

In the following example, A and B are inverses.

 

 

To find the inverse of a 2x2 matrix, use the following equation:     
 

Take the original matrixand perform the function

 

which should result in  

 

Multiply this by 1/det, where det is the determinant (Instruction for finding the determinant of a matrix can be found here: http://www.analyzemath.com/Tutorial-System-Equations/determinants.html ) 

 

 

 

Thankfully, we don't have to work out the inverses of 3x3 matrices on paper!

     1) Go to the matrix menu on your calculator and then to the edit section, where you should insert your matrix.

     2) Quit that screen and go to the clear "main menu" where you do normal calculations. Go back to the matrix

          menu and select the matrix that you edited.

     3) When the matrix is indicated by [A or whatever letter] on the screen, push the inverse button (x-11)

          and hit enter.

     4) The matrix that results is the inverse of your original!

 

 

 

Guess What? You can actually apply the skill of finding inverse matrices to another topic of Precalculus!

First of all, think about this concept.

Take Ax=B

Normally, you would divide B/A to solve for x. Unfortunately, you can't divide with matrices.

Your other option is multiplying B by the inverse of A (1/A)!

 

Let's apply this to matrices now.

 

Click Here for a video that explains and works out some inverse matrices problems.

 

 

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