How to use the MINVERSE function

What is the MINVERSE function?

The MINVERSE function calculates the inverse matrix for a given array.

What is a matrix?

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It has a fixed number of rows and columns, elements are accessed by their row and column index,

Represented by capital letters like A, B, C
Basic format is:
A = [[a,b,c],
[d,e,f],
[g,h,i]]

Rows are separated by commas, columns by brackets.

What is the inverse matrix?

For a matrix A, its inverse matrix is denoted A-1. For A to have an inverse it must be a square matrix and have non-zero determinants.

Matrix inversion is a fundamental linear algebra operation. The inverse matrix has applications in solving matrix equations, finding bases, and transforming coordinates.

How to calculate the inverse of a 2x2 matrix:

A = [[a, b], [c, d]]

To find the inverse A-1:

1. Calculate the determinant of A: det(A) = ad - bc
2. Find the adjoint of A: adj(A) = [[d, -b], [-c, a]]
3. Compute the inverse: A-1 = 1/det(A) * adj(A)

For example,

B = [[1, 2], [3, 4]]

det(B) = 14 - 23 = 4-6 = -2

adj(B) = [[4, -2], [-3, 1]]

B-1 = 1/-2 * adj(B) = [[-2, 1], [3/2, -1/2]]

So the inverse of matrix B is:

B-1 = [[-2, 1], [3/2, -1/2]]

What is an adjoint?

The adjoint of a matrix, also called the adjugate matrix, and is useful for finding the inverse of a square matrix.

To get the adjoint of an n x n matrix A:

1. Calculate the matrix of cofactors of A, denoted C.
2. Take the transpose of C to obtain the adjoint matrix, denoted adj(A).

The cofactor matrix C is obtained by replacing each element of A with its cofactor, which involves cross products of matrix minors.

Given the matrix:

A = [[a, b], [c, d]]

The cofactor matrix is:

C = [[d, -b], [-c, a]]

To get the adjoint, we take the transpose of C:

adj(A) = C^T = [[d, -c], [-b, a]]

For example:

B = [[1, 2], [3, 4]]

Cofactor matrix:

C = [[4, -2], [-3, 1]]

Adjoint:

adj(B) = [[4, -3], [-2, 1]]

What is the cofactor matrix?

What is an identity matrix?

An identity matrix is a square matrix with 1s on the main diagonal and 0s elsewhere. It is often denoted by the capital letter I.

For example, a 2x2 identity matrix:

I = [[1, 0],
[0, 1]]

Identity matrices are fundamental to linear algebra and matrix operations. Excel has a dedicated function named MUNIT function that creates an identity matrix of a given dimension.

What is a determinant?

The determinant is a special scalar value computed from a square matrix that provides crucial information about matrix properties and transformations.

Denoted as det(A) or |A| for a matrix A.

Given the matrix:

A = [[a, b], [c, d]]

The determinant is calculated as:

det(A) = ad - bc

For example:

B = [[3, 2], [1, 4]]

det(B) = (3)(4) - (2)(1) = 12 - 2 = 10

What is the MINVERSE function useful for?

This function is useful for solving equations with multiple variables.

Excel Function Syntax

MINVERSE(array)

Arguments

Comments

The MINVERSE function returns #VALUE if:

• the argument contains text or blanks.
• the number of rows and columns don't match.

MINVERSE function example

Array formula in cell B5:C6:

The MINVERSE function must be entered as an array formula.

To enter an array formula, type the formula in a cell range then press and hold CTRL + SHIFT simultaneously, now press Enter once. Release all keys.

The formula bar now shows the formula with a beginning and ending curly bracket telling you that you entered the formula successfully. Don't enter the curly brackets yourself.

Cell range B8:C9 and B11:C12 shows how the MINVERSE function calculates the inverse matrix.

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