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:
- 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)
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.
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. MUNIT function not working
The MUNTI function returns #VALUE if the dimension argument is smaller than or equal to 0 (zero).
Functions in 'Math and trigonometry' category
The MUNIT function function is one of 61 functions in the 'Math and trigonometry' category.
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