• The inverse of a matrix is used in substitution for dividing. Matrices cannot be divided but they can be multiplied by an inverse.
• Only square matrices have inverses.
• The product of a matrix and its inverse will be an identity matrix of the same dimensions where all of the values are zero excluding a diagonal line of ones. It looks similar to this:

  

If , find  

The following are the steps than can be taken to change a matrix into reduced row echelon form (rref):

• Multiply a row by a scalar quantity.
• Switching one row’s position with another.
• Adding one multiple of a row to another multiple of a row.

Create an augmented matrix with the original matrix on the left and an identity matrix on the right. The dotted line is just there to distinguish between the different sides.

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1. Press |ON|
   



2. Press |2ND| and | | to access the matrix menu.
   



3. Arrow to “EDIT” tab.
   



4. Highlight [A] and press |ENTER|.
   



5. Type “3”, |ENTER|, “3”, |ENTER|. This will set the dimensions of the matrix.
   



6. You are now ready to insert the entries of matrix O from the previous example. Type each value starting from the top-left working right, pressing |ENTER| will tab to the next position.
   



7. Press |2ND| |MODE| to return to the home screen.
   



8. Press |2ND| and | | to access the matrix menu.
   



9. [A] is highlighted, press |ENTER| to select it.
   



10. Press | | for the inverse function.
    



11. Press |ENTER| for it to calculate.
    



12. Use the arrow keys to move right and see the other values in the matrix.
    



13. To express answers in fractional form, press |MATH|.
    



14. Press |ENTER|.
    



15. Press |ENTER| for it to calculate.