Inverse Matrix in Excel

Excel Inverse Matrix (

An inverse matrix is defined as the reciprocal of a square matrix that is a non-singular matrix or invertible matrix (determinant is not equal to zero). It is hard to determine the inverse for a singular matrix. The inverse matrix in excel has an equal number of rows and columns to the original matrix.

One interesting thing about the inverse matrix is that multiplying it with the original matrix, we will get the identity matrix that has all diagonal values equal to one. Inverse matrices are applied in the linear algebra in solving the equations. To determine the inverse of a matrix, different types of methods are available including manual calculation and automated calculation. The automated calculation involves the use of Excel formulae and functions. Excel, the matrix inverse calculation process is simplified by applying an inbuilt function of MINVERSE

How to Inverse a Matrix in Excel?

Excel MINVERSE function is helpful in returning the array or matrix inverse. The input matrix must be a square matrix with all numeric values with an equal number of columns and rows in size. The INVERSE matrix will have the same dimensions as the input matrix.

Purpose: The purpose of this function is to find out the inverse of a given array

Return value: This function returns the inverse matrix with equal dimensions

Syntax: The syntax of MINVERSE function is

Array: The array should consist of only either positive or negative numerical values.

The INVERSE function is used in two ways in Excel including typing manually and inserting from the Math and Trig functions under the “Formula” tab

Uses

The inverse matrix in Excel is used for various purposes. Those include

• System of linear equations are solved in Excel using the inverse matrix
• Inverse matrices are used in non-linear equations, linear programming, and finding the integer solutions to system equations
• Inverse matrices have applications in the data analysis especially in the least square analysis to determine the various statistical parameters and values of variances and covariance’s
• Resolving problems associated with the input and output analysis in economics and business, inverse matrices are used

Examples

Example#1

Determining the inverse of a 2×2 square matrix in Excel

For this example, consider the following matrix A.

Step 1: Enter the matrix A into the Excel sheet as shown in the below-mentioned figure

The range of the matrix is that B2: C3

Step 2: Select the range of cells to position the inverse matrix A^-1 in the same sheet.

Step 3: After selecting the required cells, enter the MINVERSE function formula into the formula bar. It needs to be ensured that the formula entered while the cells are still selected.

Step 4: Enter the range of the array or matrix as shown in the screenshot.

Step 5: After entering the formula, press the ENTER key in combination with CTRL and SHIFT key to convert the normal formula to array formula to produce all elements of the inverse matrix at a time. The formula will be changed as {=MINVERSE (B2:C3)}

Step 6: The resultant inverse matrix is produced as:

Here, one thing we can observe is that the size of the input matrix and the inverse matrix is the same as 2×2.

Example #2

Determining the inverse of a 3×3 square matrix in Excel

For this example, consider the following matrix A.

Step 1: Enter the matrix A into the Excel sheet as shown in the below-mentioned figure

The range of the matrix is that B2: D4

Step 2: Select the range of cells to position the inverse matrix A^-1 in the same sheet.

Step 3: After selecting the required cells, enter the MINVERSE function formula into the formula bar. It needs to be ensured that the formula entered while the cells are still selected.

Step 4: Enter the range of the array or matrix as shown in the screenshot.

Step 5: After entering the formula, press the ENTER key in combination with CTRL and SHIFT key to convert the normal formula to array formula to produce all elements of the inverse matrix at a time. The formula will be changed as {=MINVERSE (B2: D4)}

Step 6: The resultant inverse matrix is produced as:

Here, one thing we can observe is that the size of the input matrix and the inverse matrix is the same as 3×3.

Example#3

Determining the inverse of the Identity matrix

Consider the 2×2 identity matrix for this example.

Step 1: Enter the matrix I into the Excel sheet

Step 2: Select the range of cells to position the inverse matrix I^-1 in the same sheet.

Step 3: After selecting the required cells, enter the MINVERSE function formula into the formula bar.

Step 4: Enter the range of the array or matrix as shown in the screenshot.

Step 5: Press the ENTER key in combination with CTRL and SHIFT key to convert the normal formula to array formula. The formula will be changed as {=MINVERSE (B2:C3)}

Step 6: The resultant inverse matrix is produced as:

From this, it is observed that inverse of an identity matrix and identity matrix are the same

Things to Remember

• While using the MINVERSE function in Excel, #value error will occur if the matrix contained non-numerical values, empty cells, and having a different number of columns and rows
• #NUM error is displayed if the provided matrix is a singular matrix
• #N/A error is displayed if the cells of the resulting inverse matrix are out of range. MINVERSE function results in the #N/A error in extra cells selected
• The MINVERSE function must be entered as the array formula into the spreadsheet

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