Translations

Code	Language		Translator	Run

Credits

weelookang@gmail.com; Francisco Esquembre; Felix J. Garcia Clemente

Executive Summary:

This briefing document reviews two closely related resources concerning the calculation of the determinant of N by N matrices using JavaScript. Both sources originate from the Open Educational Resources / Open Source Physics @ Singapore initiative and share authorship. The primary focus is on explaining and providing JavaScript code for calculating determinants of 2x2, 3x3, and larger square matrices using the method of Laplace expansion (cofactor expansion). The resources also highlight the use of an interactive simulation for this mathematical concept.

Main Themes and Important Ideas:

1. Introduction to Determinants: The resources introduce the concept of the determinant of a square matrix, emphasizing its importance in mathematics and linear algebra.
2. Laplace Expansion: Both sources heavily focus on the Laplace expansion method as a way to calculate the determinant of a matrix. This method involves recursively breaking down a larger matrix into smaller sub-matrices (minors) and using their determinants along with cofactors to find the determinant of the original matrix. The "Determinant of N by N Matrix JavaScript" resource explicitly states that the "Sample Learning Goals" include "Determinant of a matrix in JavaScript using Laplace expansion."
3. Step-by-Step Calculation for Small Matrices: The resources provide clear, step-by-step procedures and corresponding JavaScript code for calculating the determinant of 2x2 and 3x3 matrices.

• 2x2 Matrix: For a matrix M = [[a, bhttps://iwant2study.org/lookangejss/math/ejss_model_DeterminantNbyNMatrix/DeterminantNbyNMatrix_Simulation.xhtml " frameborder="0"></iframe>) suggests the existence of an interactive JavaScript simulation that visually demonstrates the calculation of the determinant, likely using the Laplace expansion.
• Licensing and Authorship: Both sources clearly state the authors and the Creative Commons Attribution-Share Alike 4.0 Singapore License under which the content is released. They also provide contact information for commercial use of the underlying EasyJavaScriptSimulations Library.

Key Facts and Figures:

• The resources provide JavaScript code to calculate the determinant of 2x2, 3x3, and N x N square matrices.
• The primary algorithm used is Laplace expansion (cofactor expansion).
• The code for N x N matrices is implemented recursively.
• The resources acknowledge the inefficiency of this method for large matrices.
• An interactive simulation model for visualizing the determinant calculation is available.

Quotes:

• Determinant of 2x2 matrix: "det ( M ) = ( a * d )-( b * c );"
• Determinant of 3x3 matrix: "det ( M ) = a ( ei - fh )- b ( di - fg )+ c ( dh - eg ); // notice the alternating signs"
• Inefficiency for large matrices: "This method is also very inefficient for matrices of large size. There are much more efficient ways of calculating the determinant of large square matrices."
• Laplace Expansion Formula (For Teachers): "( |A| = \sum_{j=1}^{m}(-1)^{1+j}a_{1j}|A_{1j}| )"
• Recursive det(M) function (base case for 2x2): "if ( M . length == 2 ) { return ( M [ 0 ][ 0 ]* M [ 1 ][ 1 ])-( M [ 0 ][ 1 ]* M [ 1 ][ 0 ]); }"
• Recursive det(M) function (general case): "answer += Math . pow (- 1 , i )* M [ 0 ][ i ]* det ( deleteRowAndColumn ( M , i ));"

Conclusion:

The provided resources offer a valuable educational tool for understanding and implementing the calculation of matrix determinants in JavaScript. They clearly explain the Laplace expansion method and provide functional code that works for matrices of any size, although its limitations for large matrices are noted. The inclusion of an embeddable simulation further enhances the learning experience by providing an interactive way to visualize this algebraic concept. The resources are well-attributed and licensed under an open license, making them suitable for educational use and adaptation.

Study Guide: Determinants of N x N Matrices

Key Concepts

• Square Matrix: A matrix with an equal number of rows and columns. Only square matrices have determinants.
• Determinant (det(M) or |M|): A scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether it is invertible.
• 2x2 Matrix Determinant: For a matrix ( M = $$\begin{bmatrix} a & b \ c & d \end{bmatrix}$$ ), the determinant is calculated as ( ad - bc ).
• 3x3 Matrix Determinant: For a matrix ( M = $$\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}$$ ), the determinant is calculated as ( a(ei - fh) - b(di - fg) + c(dh - eg) ).
• Laplace Expansion (Cofactor Expansion): A method for calculating the determinant of a square matrix by expanding along any row or column. The determinant is the sum of the products of the elements in that row or column and their corresponding cofactors.
• Minor: The determinant of the sub-matrix obtained by removing a particular row and column of the original matrix. For an element ( a_{ij} ), the minor ( M_{ij} ) is the determinant of the matrix formed by deleting the (i)-th row and (j)-th column.
• Cofactor: A signed minor, calculated as ( C_{ij} = (-1)^{i+j} M_{ij} ), where ( M_{ij} ) is the minor of the element in the (i)-th row and (j)-th column.
• Recursive Algorithm: A method of solving a problem where the solution depends on solutions to smaller instances of the same problem. The determinant calculation using Laplace expansion is naturally recursive.
• Base Case: In a recursive algorithm, the condition under which the recursion stops. For the determinant calculation described, the base case is the determinant of a 1x1 or 2x2 matrix.
• Efficiency: For larger matrices, Laplace expansion becomes computationally expensive. More efficient methods exist for calculating determinants of large matrices.

Quiz

1. What is the fundamental property of a matrix that allows for the calculation of its determinant? Briefly explain the formula for calculating the determinant of a 2x2 matrix.
2. Describe in your own words the process of calculating the determinant of a 3x3 matrix using the method of Laplace expansion, focusing on the first row. What pattern do you observe in the signs of the terms?
3. Explain the concept of a minor of a matrix element. How does the minor of an element ( a_{ij} ) relate to the original matrix?
4. Define the term "cofactor" of a matrix element. What is the key difference between a minor and a cofactor?
5. Describe the recursive nature of calculating the determinant using Laplace expansion. What serves as the base case for this recursion in the provided JavaScript code?
6. Explain the purpose of the deleteRowAndColumn function in the provided JavaScript code. How does it contribute to the overall determinant calculation?
7. Why is the Laplace expansion method considered inefficient for very large matrices? Briefly suggest if there might be more efficient approaches.
8. In the given JavaScript code for the general determinant function, what role does Math.pow(-1, i) play in the calculation? Why is this term necessary?
9. According to the "For Teachers" section, write down the general formula for the determinant ( |A| ) using Laplace expansion along the first row. Clearly identify each component of the formula.
10. How is the determinant of a 1x1 matrix defined according to the "Algorithm" section provided for teachers? Why is this definition important for the recursive calculation?

Quiz Answer Key

1. Only a square matrix, which has an equal number of rows and columns, has a determinant. For a 2x2 matrix ( M = $$\begin{bmatrix} a & b \ c & d \end{bmatrix}$$ ), the determinant is calculated by multiplying the diagonal elements and subtracting the product of the off-diagonal elements: ( ad - bc ).
2. For a 3x3 matrix, using Laplace expansion along the first row, we multiply each element of the first row by the determinant of the 2x2 sub-matrix obtained by removing its row and column. The signs of these terms alternate, starting with a positive sign for the first element.
3. The minor of a matrix element ( a_{ij} ) is the determinant of the smaller square matrix formed by removing the (i)-th row and the (j)-th column of the original matrix. It represents the "contribution" of the remaining elements when considering the influence of ( a_{ij} ) on the overall determinant.
4. The cofactor of a matrix element ( a_{ij} ) is its minor ( M_{ij} ) multiplied by a sign factor of ( (-1)^{i+j} ). The cofactor incorporates the alternating sign pattern used in the Laplace expansion.
5. Laplace expansion calculates the determinant of an N x N matrix by expressing it in terms of determinants of (N-1) x (N-1) sub-matrices, and this process continues recursively until it reaches smaller matrices. The base case in the JavaScript code is when the matrix is 2x2, for which the determinant is calculated directly.
6. The deleteRowAndColumn function takes a matrix and an index, and it returns a new matrix formed by removing the first row and the column at the specified index from the original matrix. This function is crucial for creating the sub-matrices needed for the recursive Laplace expansion.
7. Laplace expansion requires calculating the determinants of several sub-matrices, and the number of these calculations grows very rapidly with the size of the matrix (factorially). For large matrices, methods like Gaussian elimination to transform the matrix into an upper triangular form and then multiplying the diagonal entries are significantly more efficient.
8. The term Math.pow(-1, i) determines the sign (+ or -) of each term in the Laplace expansion along the first row. The index i represents the column number of the element being considered (starting from 0), and the alternating sign pattern is essential for the correct calculation of the determinant.
9. The general formula for the determinant ( |A| ) using Laplace expansion along the first row is: ( |A| = \sum_{j=1}^{m}(-1)^{1+j}a_{1j}|A_{1j}| ). Here, ( a_{1j} ) is the element in the first row and (j)-th column, and ( |A_{1j}| ) is the determinant of the sub-matrix obtained by removing the first row and the (j)-th column of (A).
10. The determinant of a 1x1 matrix ( |A| ) is defined as the value of its single element. This serves as the ultimate base case for the recursive definition of the determinant, as the process of Laplace expansion eventually reduces the problem to calculating determinants of 1x1 matrices.

Essay Format Questions

1. Discuss the Laplace expansion method for calculating the determinant of a square matrix. Explain the concepts of minors and cofactors and how they are used in this method. Furthermore, analyze why this method, while conceptually straightforward, becomes computationally expensive for large matrices.
2. The provided source material includes JavaScript code implementing the determinant calculation. Describe the logic behind the recursive det(M) function and the helper function deleteRowAndColumn(M, index). Explain how these functions work together to compute the determinant of an N x N matrix.
3. Compare and contrast the methods for calculating the determinant of 2x2 and 3x3 matrices as presented in the source. How does the complexity of the calculation increase from a 2x2 to a 3x3 matrix? How does Laplace expansion generalize this process for N x N matrices?
4. The source mentions that Laplace expansion is inefficient for large matrices and that more efficient methods exist. Research and discuss one such more efficient method (e.g., Gaussian elimination). Explain the fundamental principles behind this alternative method and why it offers better performance for large matrices.
5. Discuss the significance and applications of the determinant of a matrix in mathematics, computer science, or other related fields. Provide specific examples of how the determinant is used to solve problems or provide insights.

Glossary of Key Terms

• Determinant: A scalar value associated with a square matrix that can be computed from its elements. It reveals properties of the matrix, such as invertibility.
• Square Matrix: A matrix with the same number of rows and columns (n x n).
• Laplace Expansion: A method for computing the determinant of a matrix by expanding along a row or column using minors and cofactors.
• Minor (of an element): The determinant of the sub-matrix formed by deleting the row and column containing that element.
• Cofactor (of an element): The minor of that element multiplied by ( (-1)^{i+j} ), where ( i ) and ( j ) are the row and column indices of the element.
• Recursion: A programming technique where a function calls itself within its definition to solve smaller subproblems until a base case is reached.
• Base Case (in recursion): The terminating condition in a recursive function that stops the self-calling process.
• Sub-matrix: A matrix formed by deleting certain rows and/or columns from a larger matrix.
• Algorithm: A step-by-step procedure for solving a problem.
• Efficiency (of an algorithm): A measure of the computational resources (such as time and memory) required by an algorithm to solve a problem, often considered in relation to the input size.

Sample Learning Goals

Determinant of a matrix in JavaScript using Laplace expansion

https://coderbyte.com/tutorial/determinant-of-a-matrix-in-javascript-using-laplace-expansion 

Determinant of 2x2 matrix

mrdaniel will be showing you how to write a function to calculate the determinant of a square matrix of any size. Let M represent the following 2x2 matrix.

var M = [ [a,b],
          [c,d] ];

The following is the procedure to calculate the determinant of a 2x2 matrix.

det(M) = (a*d)-(b*c);

Now, we'll substitute the following values for the variables and write the basic det() function to calculate the determinant of a 2x2 matrix.

var M = [ [1,2],
          [3,4] ];

function det(M) {
   return (M[0][0]*M[1][1])-(M[0][1]*M[1][0]);
}

// output = -2 

Determinant of 3x3 matrix

The determinant of a 3x3 matrix becomes a bit trickier now. Let M be the following.

var M = [ [a,b,c],
          [d,e,f],
          [g,h,i] ];

The following is the procedure to calculate the determinant of a 3x3 matrix.

det(M) = a(ei-fh)-b(di-fg)+c(dh-eg); // notice the alternating signs

The algorithm for calculating the determinant of a 3x3 matrix is essentially the following: (1) M[0][0] * determinant of the 2x2 matrix that is excluded from M[0][0]'s row and column. (2) M[0][1] * determinant of the 2x2 matrix that is excluded from M[0][1]'s row and column. (3) M[0][2] * determinant of the 2x2 matrix that is excluded from M[0][2]'s row and column. The algorithm I'm explaining is called Laplace expansion. It technically doesn't need to take the values from the first row of the matrix and multiply them by the remainder matrices-it can actually work if you use any row. This method is also very inefficient for matrices of large size. There are much more efficient ways of calculating the determinant of large square matrices. All of this is explained in the link above. So now to update our det() function to work with 3x3 matrices. We'll rewrite the function to work recursively and our base case will be if M is a 2x2 matrix, which in that case is very simple to calculate the determinant. We'll also need to write a function that deletes a specific row and column from a matrix in order to continue with the aforementioned algorithm, and for this we'll use the JavaScript splice function.

var M = [ [1,2,3],
          [4,5,6],
          [7,8,9] ];

function det(M) {
   if (M.length==2) { return (M[0][0]*M[1][1])-(M[0][1]*M[1][0]); }
   return M[0][0]*det(deleteRowAndColumn(M,0)) - 
          M[0][1]*det(deleteRowAndColumn(M,1)) +
          M[0][2]*det(deleteRowAndColumn(M,2));
}

function deleteRowAndColumn(M,index) {
   var temp = [];
   // copy the array first
   for (var i=0; i<M.length; i++) { temp.push(M[i].slice(0)); } 
   // delete the first row
   temp.splice(0,1); 
   // delete the column at the index specified
   for (var i=0; i<temp.length; i++) { temp[i].splice(index,1); } 
   return temp;
}

// output = 0

Now let's just abstract the code in det(M) into a loop rather than writing out the three separate parts. The Math.pow() function is required in order to determine the sign of the respective part since we aren't explicitly writing out minus or plus.

function det(M) {
   if (M.length==2) { return (M[0][0]*M[1][1])-(M[0][1]*M[1][0]); }
   var answer = 0;
   for (var i=0; i<M.length; i++) { answer += Math.pow(-1,i)*M[0][i]*det(deleteRowAndColumn(M,i)); }
   return answer;
}

Determinant of 4x4 and larger matrices

The pattern simply continues now for larger matrices. Let M be the following matrix.

var M = [ [a,b,c,d],
          [e,f,g,h],
          [i,j,k,l],
          [m,n,o,p] ];

The determinant of M will therefore be:

det(M) = a*det([[f,g,h],[j,k,l],[n,o,p]])-b*det([[e,g,h],[i,k,l],[m,o,p]])+...

So because our code from above runs recursively, it should actually work for a matrix of any size 4x4 or larger. Below is the final code to determine the determinant of any size square matrix.

var M = [ [1,2,3,4],
          [5,6,7,8],
          [9,1,2,3], 
          [4,5,9,7] ];

function det(M) {
    if (M.length==1) { return (M[0][0]);} // to cater to M[0][0] case
    if (M.length==2) { return (M[0][0]*M[1][1])-(M[0][1]*M[1][0]); }
    var answer = 0;
    for (var i=0; i< M.length; i++) { answer += Math.pow(-1,i)*M[0][i]*det(deleteRowAndColumn(M,i)); }
    return answer;
}

function deleteRowAndColumn(M,index) {
    var temp = [];
    for (var i=0; i<M.length; i++) { temp.push(M[i].slice(0)); } 
    temp.splice(0,1); 
    for (var i=0; i<temp.length; i++) { temp[i].splice(index,1); } 
    return temp;
}

For Teachers

$|A| = \sum_{j=1}^{m}(-1)^{1+j}a_{1j}|A_{1j}|$

Research

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Video

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 Version:

1. https://coderbyte.com/tutorial/determinant-of-a-matrix-in-javascript-using-laplace-expansion
2. http://weelookang.blogspot.com/2018/07/determinant-of-n-by-n-matrix-javascript.html 

Other Resources

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