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 IIt 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 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 eq=\frac{1}{det}, where det is the determinant (Instructions for finding the determinant of a matrix can be found here: The-Determinant-of-a-Square-Matrix





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 (eq=x^-1)

          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 : eq=\frac{1}{a} !


Now, let's use this idea to solve systems of equations with matrices.

     example system:

          4x + 2y - 2z = 10

          2x + 8y + 4z = 32

          30x + 12y - 4z = 24


1) Separate the elements into three matrices: a coefficient matrix, a variable matrix (with x,y,z,etc), and a matrix with the numbers on the opposite side of the equal sign.


2) Use your calculator (!) and the skills that we have reviewed to solve!

     ~Insert the Coefficient matrix as [A] and the final 3x1 matrix as [B].

     ~Then, just type in [A] inverse multiplied by [B].

     ~The three numbers in the resulting matrix are your solutions in the order that you inserted your

          variables into the variable matrix.

Solution to our example: (-2, 6, -3)


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