The inverse of a square matrix is the matrix with the same dimensions that, when multiplied with the original matrix, gives the Identity Matrix:
If an inverse exists, the original matrix is known as invertible. Otherwise, the orginal matrix is described as singular.
Further details of the Matrix Inversion are given on WikipediaThe Excel MINVERSE function calculates the inverse of a square matrix.
The syntax of the function is:
where the array argument is an array of values representing a square matrix (i.e. a matrix that has the same number of rows as columns).
The resulting inverse matrix has the same number of rows and columns as the original supplied array. Be aware, however, that the result from the Minverse function result is calculated to approximately 16 decimal places, so you may get some small rounding errors.
Note: as the Excel Minverse function returns an array of values, the function must be entered into your Worksheet as an Array Formula.
Inputting Array Formulas
To input an array formula, you need to first highlight the range of cells that the function is to be entered into. Type the function into the first cell of the range, and press CTRLSHIFTEnter.
Go to the Excel Array Formulas page for more details.
In the example below, the Excel Minverse function is used to find the inverse of the 4x4 matrix in cells A1D4 of the example spreadsheet. The Minverse function is entered into cells F1I4 of the spreadsheet.
The formula can be seen in the formula bar of the result spreadsheet. The curly braces { } show that it has been input as an Array Formula.
Original Matrix and Formula: 
Result (formula entered as an array formula): 
We can test the Minverse function by multiplying the two matrices in the above spreadsheet, using the Mmult Function. This gives the 4x4 Identity Matrix:
For further information and examples of the Excel Minverse function, see the Microsoft Office website.
If you get an error from the Excel Minverse function this is likely to be one of the following:
#VALUE!    Occurs if either:

#NUM!    Returned if the supplied matrix is singular  i.e. there is no inverse for the supplied matrix. 
#N/A    Occurs in cells outside the range of the resulting matrix. For example, if, in the example above, we had highlighted cells F1I5 before entering the Minverse function, the 'additional' cells, F5I5 are not part of the resulting matrix. Therefore, these cells will display the #N/A error. 
Be aware also that if, when entering the Minverse function, you don't highlight the entire range of cells that is needed to display the resulting matrix, Excel will just show the part of the result that fits into the highlighted range. There will be no error message to let you know that your highlighted range is not big enough. This will also be the case if you fail to enter the formula as an array formula  the cell that you enter the formula into will simply show you the first entry of the inverse matrix. 