# A to Z of Excel Functions: The MINVERSE Function

13 June 2022

*Welcome back to our regular A to Z of Excel Functions blog. Today we look at the MINVERSE function. *

**The MINVERSE function**

In mathematics, especially in areas such as linear algebra, matrices may be used to solve simultaneous equations. For the record, a matrix is not just a movie it’s a rectangular arrangement of **m **x **n** elements, in the dimensions of **m **rows by **n **columns, *e.g.* a matrix **A** *(say)* may be represented as

It is often written in compact form as

An **n **x **n** square matrix **A** is called invertible (also non-singular or non-degenerate), if there exists an **n **x **n** square matrix **B** such that

**AB = BA =I _{n}**

where **I _{n}** denotes the

**n**x

**n**identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix

**B**is uniquely determined by

**A**, and is called the (multiplicative) inverse of

**A**, denoted by

**A**. Matrix inversion is the process of finding the matrix

^{?1}**B**that satisfies the prior equation for a given invertible matrix

**A**.

A square matrix that is not invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any finite region on the number line or complex plane, the probability that the matrix is singular is zero [0], that is, it will "almost never" be singular. Non-square matrices (**m **x **n** matrices for which **m** ? **n**) do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse. If **A** is **m **x **n** and the rank of **A** is equal to **n** (**n** ? **m**), then **A** has a *left *inverse, an **n **x **m** matrix **B** such that **BA** = **I _{n}**. If

**A**has rank

**m**(

**m**?

**n**), then it has a

*right*inverse, an

**n**x

**m**matrix

**B**such that

**AB**=

**I**.

_{m}The Excel function **MINVERSE **returns the inverse matrix for a matrix stored in an array. It has the following syntax:

**MINVERSE(array)**

where:

**array**is required, and represents a numerical**array**with an equal number of rows and columns.

It should be noted that:

**array**may be given as:- a cell range,
*e.g.***A1:C3** - an
**array**constant, such as**{1,2,3;4,5,6;7,8,9}** - a name to either of these

- a cell range,
**MINVERSE**returns the*#VALUE!*error when:- any cells in array are empty or contain text
**array**does not have an equal number of rows and columns

- matrix determinants are generally used for solving systems of mathematical equations that involve several variables
**MINVERSE**is calculated with an accuracy of approximately 16 digits, which may lead to a small numeric error when the calculation is not complete. For example, the determinant of a singular matrix may differ from zero by 1E-16- some square matrices cannot be inverted and will return the
*#NUM!*error value with**MINVERSE**. The determinant for a non-invertible matrix is zero [0].

If you have a current version of Microsoft 365, then you can simply enter the formula in the top-left-cell of the output range, then press **ENTER** to confirm the formula as a dynamic array formula. Otherwise, the formula must be entered as a legacy array formula by first selecting the output range, entering the formula in the top-left-cell of the output range, and then pressing **CTRL + SHIFT + ENTER** to confirm it. Excel inserts curly brackets (“braces”) at the beginning and end of the formula.

As an example:

*We’ll continue our A to Z of Excel Functions soon. Keep checking back – there’s a new blog post every business day.*

*A full page of the function articles can be found* *here. *