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:
- Calculate the determinant of A: det(A) = ad - bc
- Find the adjoint of A: adj(A) = [[d, -b], [-c, a]]
- 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:
- Calculate the matrix of cofactors of A, denoted C.
- 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
array | Required. An array containing numbers, the number of rows and columns must be the same. |
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.
'MINVERSE' function examples
The following article has a formula that contains the MINVERSE function.
This article demonstrates how to solve simultaneous linear equations using formulas and Solver. The variables have the same value in […]
Functions in 'Math and trigonometry' category
The MINVERSE function function is one of 73 functions in the 'Math and trigonometry' category.
How to comment
How to add a formula to your comment
<code>Insert your formula here.</code>
Convert less than and larger than signs
Use html character entities instead of less than and larger than signs.
< becomes < and > becomes >
How to add VBA code to your comment
[vb 1="vbnet" language=","]
Put your VBA code here.
[/vb]
How to add a picture to your comment:
Upload picture to postimage.org or imgur
Paste image link to your comment.
Contact Oscar
You can contact me through this contact form