Cofactor Expansion Calculator

Introduction to Cofactor Expansion Calculator?

Cofactor Expansion Calculator with steps is an amazing tool that is used to compute the determinant of a matrix using the cofactor expansion method. Our tool helps you to find the cofactor expansion of determinants for different orders whether it has 2 by 2, 3 by 3, or even 4 by 4 order.

The cofactor calculator is a valuable tool for students, researchers, and professionals as It simplifies the process of evaluating the determinants to find the specific elements from matrices in linear algebra.

What is Cofactor Expansion?

Cofactor expansion is a method that is used to calculate the determinant of a square matrix or to find specific elements within the matrix. It is known as cofactor expansion or expansion by minors method.

Cofactor expansion is widely used in solving systems of linear equations, calculating the inverse of a matrix, and understanding the properties of transformations and transformations of spaces.

How to Find Determinants by Cofactor Expansion?

To find the determinant using the cofactor expansion formula, the cofactor expansion calculator breaks down the n by n order of the matrix into smaller parts as the cofactors of 2x2 matrices in which each element is associated with that matrix. Here is a step-by-step guide on how to perform cofactor expansion:

Step 1:

Identify the given matrix and its order.

$$ A \;=\; \left[ \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{matrix} \right] $$

Step 2:

To make the cofactor(breakdown into the simplest matrix), select any row or column of the matrix for expansion such as:

If you select first row then leave the first row and first column and select the leftover matrix as a cofactor.

$$ a_{11} \;=\; C_{11} \;=\; (-1)^{1+1} \left| \begin{matrix} a_{22} & a_{23} \\ a_{32} & a_{33} \\ \end{matrix} \right| \;=\; +(a_{22} . a_{33} - a_{23} . a_{32}) $$

If you select the second row then leave the second row and second column and select the leftover matrix as a cofactor.

$$ a_{32} \;=\; C_{23} \;=\; (-1)^{2+3} \left| \begin{matrix} a_{11} & a_{13} \\ a_{21} & a_{23} \\ \end{matrix} \right| \;=\; -(a_{11} . a_{23} - a_{13} . a_{21}) $$

If you select the third row then leave the third row and third column and select the leftover matrix as a cofactor.

$$ a_{23} \;=\; C_{23} \;=\; (-1)^{2+3} \left| \begin{matrix} a_{11} & a_{12} \\ a_{31} & a_{32} \\ \end{matrix} \right| \;=\; -(a_{11} . a_{32} - a_{12} . a_{31}) $$

Step 3:

For each element in the selected row or column, calculate its cofactor. The cofactor of an element is given by:

$$ C_{ij} \;=\; (−1)i + j ⋅ M_{ij} $$

Where Mij is the determinant of the cofactor matrix (submatrix) obtained by removing the i-th row and j-th column from the original matrix A.

Step 4:

Solve the cofactor expanded matrix to ith or jth form,

$$ der(A) \;=\; \sum_{j=1}^{n} a_{ij} C_{ij} $$

$$ det(A) \;=\; \sum_{i=1}^{n} a_{ij} C_{ij} $$

Solved Example of Cofactor Expansion:

Lets us see an example to understand the cofactor expansion of a determinant to understand how the cofactor expansion calculator with steps works.

Example:

$$ \left( \begin{matrix} 2 & 0 & 3 \\ 1 & -1 & 4 \\ 5 & 2 & 0 \\ \end{matrix} \right) $$

Solution:

For a11:

$$ \left( \begin{matrix} -1 & 4 \\ 2 & 0 \\ \end{matrix} \right) $$

$$ det \left( \left( \begin{matrix} -1 & 4 \\ 2 & 0 \\ \end{matrix} \right) \right) \;=\; (-1)(0) - (4)(2) \;=\; -8 $$

$$ C_{11} \;=\; (-1)^{1+1} . (-8) \;=\; -8 $$

For a12:

$$ \left( \begin{matrix} 1 & 4 \\ 5 & 0 \\ \end{matrix} \right) $$

$$ det \left( \left( \begin{matrix} 1 & 4 \\ 5 & 0 \\ \end{matrix} \right) \right) \;=\; (1)(0) - (4)(5) \;=\; -20 $$

$$ C_{12} \;=\; (-1)^{1+2} . (-20) \;=\; 20 $$

For a13:

$$ \left( \begin{matrix} 1 & -1 \\ 5 & 2 \\ \end{matrix} \right) $$

$$ det \left( \left( \begin{matrix} 1 & -1 \\ 5 & 2 \\ \end{matrix} \right) \right) \;=\; (1)(2) - (-1)(5) \;=\; 2 + 5 \;=\; 7 $$

$$ C_{13} \;=\; (-1)^{1+3} . 7 \;=\; 7 $$

Calculate the determinant of matrix A,

$$ det(A) \;=\; a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} \;=\; (2)(-8) + (0)(20) + (3)(7) $$

$$ det(A) \;=\; -16 + 0 + 21 \;=\; 5 $$

So the solution of given cofactor determinant matrix problem is 5.

How to Use Cofactor Expansion Calculator?

The cofactor calculator has a user-friendly design that enables you to use it to easily calculate cofactor expansion questions. Before adding the input as a determinant matrix for solutions, you just need to follow some simple steps. These steps are:

1. Add the size of the cofactor matrix expansion
2. Enter the elements of the matrix in the input box to find the cofactor expansion question with the solution.
3. Recheck your input matrix value before hitting the calculate button to start the calculation process in the cofactor expansion 4x4 calculator.
4. Click on the “Calculate” button to get the desired result of your given cofactor expansion questions with a solution.
5. If you want to try out our cofactor expansion determinant calculator for the first time then you can use the load example.
6. Click on the “Recalculate” button to get a new page for solving cofactor expansion problems to get solutions.

Results from Cofactor Calculator:

Cofactor Expansion Calculator with steps gives you the solution to a given problem when you add the input value in it. It provides you with the solutions of the cofactor expansion question. It may contain as:

• Result Option:

When you click on the result option then it provides you with a solution for the cofactor expansion question

• Possible Step:

When you click on the possible steps option it provides you with the solution of cofactor expansion problems in steps.

Advantages of Cofactor Expansion 4x4 Calculator:

The cofactor calculator has many advantages that you obtain whenever you use it to calculate cofactor expansion problems and get solutions without giving any manual guide. These advantages are:

• It is a free tool so you can use it to find the cofactor expansion problems with a solution.
• The cofactor expansion determinant calculator is an adaptable tool that can operate through electronic devices like laptops, computers, mobile, tablets, etc with the help of the internet only.
• Our tool saves the time and effort that you consume in doing lengthy calculations of cofactor expansion in a few seconds.
• It is a learning tool so you can use our tool for practice so that you get in-depth knowledge.
• It provides you solutions in a complete process in a step-by-step method for a better understanding of Cofactor Expansion questions.
• It is a trustworthy tool that provides you with accurate solutions according to your input whenever you use the Cofactor Expansion Calculator with steps to get a solution.