Introduction to Cofactor Matrix Calculator:
Cofactor Matrix Calculator is an online tool that helps you to find the cofactor of a given matrix in a run of time. Our tool helps you to evaluate the determinant matrix of n by n order with the help of the cofactor method from linear algebra.
The matrices cofactor calculator is a valuable tool for learning the determinant matrix problems because it can simplify the process of finding cofactors, which makes the calculation of a matrix easy and quick.
What is a Cofactor Matrix?
A cofactor matrix is a process in which the matrix each element is replaced by its corresponding cofactor in the determinant matrix to form a new matrix.
In linear algebra, if a square matrix A has order n×n, the cofactor Cij of an element aij that is multiplied with (-1)ij is defined as:
$$ Cij \;=\; (−1)i + j⋅det(Mij) $$
Where:
- i and j are the row and columns that shows of aij.
- Mij is the minor matrix that obtained from the i-th row and j-th column from A.
- det(Mij) denotes the determinant of matrix Mij.
How to Find Cofactor Matrix?
For calculating the cofactor matrix, the calculator cofactor matrix follows the given steps to solve the cofactor matrix.
- Identify the matrix minor Mij of the element aij which is obtained from i-th row and j-th column from matrix A.
- Calculate the Determinant of the minor after finding the determinant of the smaller matrix formed, after removing the specified row and column.
- Apply the Cofactor Formula, the cofactor Cij formula of the element aij used by the cofactors of matrix calculator is given by: $$ Cij \;=\; (−1)i + j⋅det(Mij) $$
- Lastly, add all the elements of cofactor into one matrix.
Solved Example of Cofactor Matrix:
Let's see an example of a cofactor matrix with the solution in which thecalculation process of the Cofactor Matrix Calculator is explained.
Example:
Find cofactor B,
$$ B \;=\; \left[ \begin{matrix} 2 & 3 & 1 \\ 0 & 5 & 6 \\ 1 & 1 & 2 \\ \end{matrix} \right] $$
Solution:
$$ Cofactor(B) \;=\; Cofactor\; \left[ \begin{matrix} 2 & 3 & 1 \\ 0 & 5 & 6 \\ 1 & 1 & 2 \\ \end{matrix} \right] $$
Let us make the cofactor of the given matrix with the help of a minor technique.
$$ Cofactor\; of\; 2 \;=\; B_{11} \;=\; + \biggr| \begin{matrix} 5 & 6 \\ 1 & 2 \\ \end{matrix} \biggr| \;=\; + (5 \times 2 - 6 \times 1) \;=\; +(10 - 6) \;=\; 4 $$
$$ Cofactor\; of\; 3 \;=\; B_{12} \;=\; - \biggr| \begin{matrix} 0 & 6 \\ 1 & 2 \\ \end{matrix} \biggr| \;=\; - (0 \times 2 - 6 \times 1) \;=\; - (0 - 6) \;=\; 6 $$
$$ Cofactor\; of\; 1 \;=\; B_{13} \;=\; + \biggr| \begin{matrix} 0 & 5 \\ 1 & 1 \\ \end{matrix} \biggr| \;=\; + (0 \times 1 - 5 \times 1) \;=\; + (0 - 5) \;=\; -5 $$
$$ Cofactor\; of\; 0 \;=\; B_{21} \;=\; - \biggr| \begin{matrix} 3 & 1 \\ 1 & 2 \\ \end{matrix} \biggr| \;=\; - (3 \times 2 - 1 \times 1) \;=\; - (6 - 1) \;=\; -5 $$
$$ Cofactor\; of\; 5 \;=\; B_{22} \;=\; + \biggr| \begin{matrix} 2 & 1 \\ 1 & 2 \\ \end{matrix} \biggr| \;=\; + (2 \times 2 - 1 \times 1) \;=\; + (4 - 1) \;=\; 3 $$
$$ Cofactor\; of\; 6 \;=\; B_{23} \;=\; - \biggr| \begin{matrix} 2 & 3 \\ 1 & 1 \\ \end{matrix} \biggr| \;=\; - (2 \times 1 - 3 \times 1) \;=\; - (2 - 3) \;=\; 1 $$
$$ Cofactor\; of\; 1 \;=\; B_{31} \;=\; + \biggr| \begin{matrix} 3 & 1 \\ 5 & 6 \\ \end{matrix} \biggr| \;=\; + (3 \times 6 - 1 \times 5) \;=\; + (18 - 5) \;=\; 13 $$
$$ Cofactor\; of\; 1 \;=\; B_{32} \;=\; - \biggr| \begin{matrix} 2 & 1 \\ 0 & 6 \\ \end{matrix} \biggr| \;=\; - (2 \times 6 - 1 \times 0) \;=\; - (12 + 0) \;=\; -12 $$
$$ Cofactor\; of\; 2 \;=\; B_{33} \;=\; + \biggr| \begin{matrix} 2 & 3 \\ 0 & 5 \\ \end{matrix} \biggr| \;=\; + (2 \times 5 - 3 \times 0) \;=\; + (10 + 0) \;=\; 10 $$
After finding all the cofactors, let's combine all these cofactors in a matrix B such as:
$$ \left[ \begin{matrix} B_{11} & B_{12} & B_{13} \\ B_{21} & B_{22} & B_{23} \\ B_{31} & B_{32} & B_{33} \\ \end{matrix} \right] \;=\; \left[ \begin{matrix} 4 & 6 & -5 \\ -5 & 3 & 1 \\ 13 & -12 & 10 \\ \end{matrix} \right] $$
How to Use Cofactor Matrix Calculator?
The matrices cofactor calculator has a simple layout that helps you to find the cofactor matrix from the given determinant matrix instantly.
You just need to put your problem in this calculator cofactor matrix only by following our guidelines that keep you away from trouble. These guidelines are:
- Choose the size of the matrix as per your cofactor matrix problem
- Enter the elements of the matrix that you want to evaluate for the cofactor matrix in the input field
- Review your given input value before clicking the calculate button to get the exact solution of the cofactor matrix
- Click the “Calculate” button for the solution of cofactor matrix problems
- If you want to check the cofactors of matrix calculator then use the load example for the calculation.
- Click the “Recalculate” button for the solution of more examples of the cofactor matrix question
Final Result of Matrices Cofactor Calculator:
Cofactor Matrix Calculator provides you solution for finding the cofactor matrix from a given matrix as per your input values when you click on the calculate button. It may include as:
In the Result Box:
When you click on the result button you get the solution in the form of a new matrix
Steps Box:
Click on the steps option so that you get the solution of cofactor matrix questions in a step-by-step method.
Advantages of Calculator Cofactor Matrix:
The matrix cofactor calculator has millions of advantages when you use it to solve the matrix problems for finding the cofactor matrix in the solution. You just need to give the input value and get a solution without imposing any restrictions.
- The matrices cofactor calculator is a trustworthy tool that always provides you with accurate solutions of given matrix cofactor problem.
- It is an efficient tool that evaluates cofactor matrix problems with solutions in a run-of-time.
- The cofactors of matrix calculator is a learning tool that helps children to learn about the concept of cofactor matrix very easily on online platforms.
- It is a handy tool that can solve different orders of matrix to find the cofactor matrix quickly without any external effort.
- It is a free tool that allows you to use it for the calculation of cofactor matrix problems.
- It is an easy-to-use tool, anyone or even a beginner can easily use it for the solution of cofactor matrix problems.
- Cofactor Matrix Calculator can be used on all devices like desktops, mobile, or laptops on the internet to get solutions of cofactor matrix problems.
