How to use the MUNIT function

What is the MUNIT function?

The MUNIT function calculates the identity matrix for a given dimension.

1. Introduction

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.

What applications does the identity matrix have?

• Multiplying by the identity matrix leaves the original matrix unchanged.
• Finding the inverse of a matrix involves multiplying by the identity matrix.

Explain how multiplying by the identity matrix returns the original matrix?

Let's say we have the matrix:

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

And the 2x2 identity matrix:

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

When we multiply A and I:

A x I = [[1, 2], x [[1, 0], [3, 4]] [0, 1]]

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

What is the inverse of a 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)

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.

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 are the matrix functions in Excel?

Function	Description
MMULT(array1, array2)	Returns the matrix product of two arrays.
MUNIT(dimension)	Returns the matrix identity for the specified dimension.
MINVERSE(array)	Returns the matrix inverse of the given array.
MDETERM(array)	Returns the matrix determinant of the given array.

2. Syntax

MUNIT(dimension)

dimension

Required. An integer that determines the dimension of the returning unit matrix.

3. Example 1

This example shows how to create an identity matrix based on a given number, in this case 3, that represents the dimension.

Array formula in cell B3:

The formula above returns an identity matrix with dimension 3, meaning the matrix is 3x3 and has ones on the main diagonal and zeros elsewhere.

The MUNIT function returns an array, you need to enter the function as an array formula. However, Excel 365 subscribers may enter the formula as a regular formula. Skip the steps below.

To enter an array formula, type the formula in a cell 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.

The dimension argument must be larger than 0 (zero).

4. Example 2

Multiply a 3x5 matrix by the 5x5 identity matrix to verify that the identity matrix does not change the original matrix when multiplied?

The image above demonstrates how to multiply a matrix with 3 rows and 5 columns by an identity matrix with 5 dimensions. The following table describes the values of the 3x5 matrix:

	B	C	D	E	F
3	81	42	20	81	85
4	88	39	47	33	40
5	69	62	4	27	22

The MUNIT argument is:

• dimension: 5

Formula in cell B8:

The formula in cell B8 returns the exact same matrix (3x5) as the original matrix.

	B	C	D	E	F
8	81	42	20	81	85
9	88	39	47	33	40
10	69	62	4	27	22

The image above shows the original 3x5 matrix in cell range B3:F5 and the result of multiplying the original 3x5 matrix with a identity matrix with 5 dimensions in cell range B8:F10.

5. Example 3

Calculate the determinant of a 3x3 identity matrix?

The image above shows an 3x3 identity matrix in cell range B3:D5.

Formula in cell C7:

The formula in cell C7 returns 1 which represents the determinant of a 3x3 identity matrix. In fact, any size of identity matrix returns 1.

This means that if you calculate the determinant of an identity matrix using the MDETERM function in Excel, the result will be 1, regardless of the dimension n.

Explaining formula in cell C7

Step 1 - Create an identity matrix with 3 rows and 3 columns

MUNIT(3)

returns

1	0	0
0	1	0
0	0	1

Step 2 - Calculate the determinant of an 3x3 identity matrix

MDETERM(MUNIT(3))

becomes

MDETERM({1,0,0;0,1,0;0,0,1})

and returns 1.

6. Example 4

Calculate the inverse of a 3x3 identity matrix?

The image above shows an 3x3 identity matrix in cell range B3:D5.

Formula in cell C7:

The formula in cell C7 returns the inverse of an 3x3 identity matrix which is equal to itself. In fact, the inverse of an identity matrix of any dimension is the identity matrix itself.

This means that if you calculate the inverse of an identity matrix using the MINVERSE function in Excel, the result will be the original identity matrix itself, regardless of the dimension n.

7. Example 5

Given a 3x3 matrix A, verify that A * A^-1 = I ?  A^-1 is the inverse matrix of A. I is the identity matrix.

Cell range B3:D5 contains matrix A, it has 3 rows and 3 columns. Cell range B8:D10 contains the inverse matrix A^-1

Formula in cell B13:

The result in cell range B13:D15 is the identity matrix based on multiplying matrix A by the inverse of matrix A^-1

Explaining formula in cell B13

Step 1 - Calculate the inverse matrix

MINVERSE(B3:D5)

Step 2 - Multiply matrix A by matrix A^-1

MMULT(B3:D5,MINVERSE(B3:D5))

8. Function not working

The MUNIT function returns

• #VALUE if the dimension argument is
  • smaller than or equal to 0 (zero).
  • is non-numeric.
• #NAME? error if you misspell the function name.
• propagates errors, meaning that if the input contains an error (e.g., #VALUE!, #REF!), the function will return the same error.

8.1 Troubleshooting the error value

When you encounter an error value in a cell a warning symbol appears, displayed in the image above. Press with mouse on it to see a pop-up menu that lets you get more information about the error.

1. The first line describes the error if you press with left mouse button on it.
2. The second line opens a pane that explains the error in greater detail.
3. The third line takes you to the "Evaluate Formula" tool, a dialog box appears allowing you to examine the formula in greater detail.
4. This line lets you ignore the error value meaning the warning icon disappears, however, the error is still in the cell.
5. The fifth line lets you edit the formula in the Formula bar.
6. The sixth line opens the Excel settings so you can adjust the Error Checking Options.

Here are a few of the most common Excel errors you may encounter.

#NULL error - This error occurs most often if you by mistake use a space character in a formula where it shouldn't be. Excel interprets a space character as an intersection operator. If the ranges don't intersect an #NULL error is returned. The #NULL! error occurs when a formula attempts to calculate the intersection of two ranges that do not actually intersect. This can happen when the wrong range operator is used in the formula, or when the intersection operator (represented by a space character) is used between two ranges that do not overlap. To fix this error double check that the ranges referenced in the formula that use the intersection operator actually have cells in common.

#SPILL error - The #SPILL! error occurs only in version Excel 365 and is caused by a dynamic array being to large, meaning there are cells below and/or to the right that are not empty. This prevents the dynamic array formula expanding into new empty cells.

#DIV/0 error - This error happens if you try to divide a number by 0 (zero) or a value that equates to zero which is not possible mathematically.

#VALUE error - The #VALUE error occurs when a formula has a value that is of the wrong data type. Such as text where a number is expected or when dates are evaluated as text.

#REF error - The #REF error happens when a cell reference is invalid. This can happen if a cell is deleted that is referenced by a formula.

#NAME error - The #NAME error happens if you misspelled a function or a named range.

#NUM error - The #NUM error shows up when you try to use invalid numeric values in formulas, like square root of a negative number.

#N/A error - The #N/A error happens when a value is not available for a formula or found in a given cell range, for example in the VLOOKUP or MATCH functions.

#GETTING_DATA error - The #GETTING_DATA error shows while external sources are loading, this can indicate a delay in fetching the data or that the external source is unavailable right now.

8.2 The formula returns an unexpected value

To understand why a formula returns an unexpected value we need to examine the calculations steps in detail. Luckily, Excel has a tool that is really handy in these situations. Here is how to troubleshoot a formula:

1. Select the cell containing the formula you want to examine in detail.
2. Go to tab “Formulas” on the ribbon.
3. Press with left mouse button on "Evaluate Formula" button. A dialog box appears.
   The formula appears in a white field inside the dialog box. Underlined expressions are calculations being processed in the next step. The italicized expression is the most recent result. The buttons at the bottom of the dialog box allows you to evaluate the formula in smaller calculations which you control.
4. Press with left mouse button on the "Evaluate" button located at the bottom of the dialog box to process the underlined expression.
5. Repeat pressing the "Evaluate" button until you have seen all calculations step by step. This allows you to examine the formula in greater detail and hopefully find the culprit.
6. Press "Close" button to dismiss the dialog box.

There is also another way to debug formulas using the function key F9. F9 is especially useful if you have a feeling that a specific part of the formula is the issue, this makes it faster than the "Evaluate Formula" tool since you don't need to go through all calculations to find the issue..

1. Enter Edit mode: Double-press with left mouse button on the cell or press F2 to enter Edit mode for the formula.
2. Select part of the formula: Highlight the specific part of the formula you want to evaluate. You can select and evaluate any part of the formula that could work as a standalone formula.
3. Press F9: This will calculate and display the result of just that selected portion.
4. Evaluate step-by-step: You can select and evaluate different parts of the formula to see intermediate results.
5. Check for errors: This allows you to pinpoint which part of a complex formula may be causing an error.

The image above shows cell reference B3 converted to hard-coded value using the F9 key. The MUNIT function requires numerical values which is not the case in this example. We have found what is wrong with the formula.

Tips!

• View actual values: Selecting a cell reference and pressing F9 will show the actual values in those cells.
• Exit safely: Press Esc to exit Edit mode without changing the formula. Don't press Enter, as that would replace the formula part with the calculated value.
• Full recalculation: Pressing F9 outside of Edit mode will recalculate all formulas in the workbook.

Remember to be careful not to accidentally overwrite parts of your formula when using F9. Always exit with Esc rather than Enter to preserve the original formula. However, if you make a mistake overwriting the formula it is not the end of the world. You can “undo” the action by pressing keyboard shortcut keys CTRL + z or pressing the “Undo” button

8.3 Other errors

Floating-point arithmetic may give inaccurate results in Excel - Article

Floating-point errors are usually very small, often beyond the 15th decimal place, and in most cases don't affect calculations significantly.

 

Functions in 'Math and trigonometry' category

The MUNIT function function is one of 62 functions in the 'Math and trigonometry' category.

Excel function categories

Excel categories

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