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Cofactor Matrix calculator takes matrix as input and finds its cofactor matrix, with detailed step by step calculations.
How to use Cofactor Matrix Calculator?
1. Find the cofactor matrix of the following matrix A.
\( A = \begin{bmatrix}1 & 2 & 3\\4 & 5 & 6\\9 & 8 & 7\end{bmatrix} \)
Solution
Since the given matrix is of size 3×3, follow these steps with the Cofactor Matrix Calculator.
- Enter value of 3 in input fields: m=3 and n=3.
- A matrix with specified size of 3×3 appears, with input field for each element in the matrix.
- Enter the given matrix values, and click on Calculate button.
- A step-by-step solution with the following steps will be displayed.
Given:
Matrix \( A = \begin{bmatrix}1 & 2 & 3\\4 & 5 & 6\\9 & 8 & 7\end{bmatrix} = \begin{bmatrix}a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33}\end{bmatrix} \)
Computing Cofactor Matrix
The formula to find the cofactor of each element \(a_{ij}\) in matrix A, where \(a_{ij}\) is the element at ith row, and jth column is
Cofactor of \(a_{ij} = (-1)^{i+j}\cdot det(minor(a_{ij}))\)
where
- \(minor(a_{ij})\) is the minor matrix of element \(a_{ij}\)
- \(det(minor(a_{ij}))\) is the determinant of minor matrix of element \(a_{ij}\)
Cofactors of elements in row 1
Cofactor of \(a_{11} = (-1)^{1+1}\cdot |minor(a_{11})|\) \( = (-1)^{2}\cdot \begin{vmatrix}5 & 6\\8 & 7\end{vmatrix} \)\( = -13 \)
Cofactor of \(a_{12} = (-1)^{1+2}\cdot |minor(a_{12})|\) \( = (-1)^{3}\cdot \begin{vmatrix}4 & 6\\9 & 7\end{vmatrix} \)\( = 26 \)
Cofactor of \(a_{13} = (-1)^{1+3}\cdot |minor(a_{13})|\) \( = (-1)^{4}\cdot \begin{vmatrix}4 & 5\\9 & 8\end{vmatrix} \)\( = -13 \)
Cofactors of elements in row 2
Cofactor of \(a_{21} = (-1)^{2+1}\cdot |minor(a_{21})|\) \( = (-1)^{3}\cdot \begin{vmatrix}2 & 3\\8 & 7\end{vmatrix} \)\( = 10 \)
Cofactor of \(a_{22} = (-1)^{2+2}\cdot |minor(a_{22})|\) \( = (-1)^{4}\cdot \begin{vmatrix}1 & 3\\9 & 7\end{vmatrix} \)\( = -20 \)
Cofactor of \(a_{23} = (-1)^{2+3}\cdot |minor(a_{23})|\) \( = (-1)^{5}\cdot \begin{vmatrix}1 & 2\\9 & 8\end{vmatrix} \)\( = 10 \)
Cofactors of elements in row 3
Cofactor of \(a_{31} = (-1)^{3+1}\cdot |minor(a_{31})|\) \( = (-1)^{4}\cdot \begin{vmatrix}2 & 3\\5 & 6\end{vmatrix} \)\( = -3 \)
Cofactor of \(a_{32} = (-1)^{3+2}\cdot |minor(a_{32})|\) \( = (-1)^{5}\cdot \begin{vmatrix}1 & 3\\4 & 6\end{vmatrix} \)\( = 6 \)
Cofactor of \(a_{33} = (-1)^{3+3}\cdot |minor(a_{33})|\) \( = (-1)^{6}\cdot \begin{vmatrix}1 & 2\\4 & 5\end{vmatrix} \)\( = -3 \)
Consolidate all the cofactors computed for elements in the matrix A to get the Cofactor Matrix of A.
Result
Therefore, the cofactor matrix of given matrix A, is
Cofactor Matrix of A \( = \begin{bmatrix}-13 & 26 & -13\\10 & -20 & 10\\-3 & 6 & -3\end{bmatrix} \)
2. Find the cofactor matrix of the following matrix A.
\( A = \begin{bmatrix}5 & 7\\3 & 1\end{bmatrix} \)
Solution
Since the given matrix is of size 2×2, follow these steps with the Cofactor Matrix Calculator.
- Enter value of 2 in input fields: m=2 and n=2.
- A matrix with specified size of 2×2 appears, with input field for each element in the matrix.
- Enter the given matrix values, and click on Calculate button.
- A step-by-step solution with the following steps will be displayed.
Given:
Matrix \( A = \begin{bmatrix}5 & 7\\3 & 1\end{bmatrix} = \begin{bmatrix}a_{11} & a_{12}\\a_{21} & a_{22}\end{bmatrix} \)
Computing Cofactor Matrix
The formula to find the cofactor of each element \(a_{ij}\) in matrix A, where \(a_{ij}\) is the element at ith row, and jth column is
Cofactor of \(a_{ij} = (-1)^{i+j}\cdot det(minor(a_{ij}))\)
where
- \(minor(a_{ij})\) is the minor matrix of element \(a_{ij}\)
- \(det(minor(a_{ij}))\) is the determinant of minor matrix of element \(a_{ij}\)
Cofactors of elements in row 1
Cofactor of \(a_{11} = (-1)^{1+1}\cdot |minor(a_{11})|\) \( = (-1)^{2}\cdot \begin{vmatrix}1\end{vmatrix} \)\( = 1 \)
Cofactor of \(a_{12} = (-1)^{1+2}\cdot |minor(a_{12})|\) \( = (-1)^{3}\cdot \begin{vmatrix}3\end{vmatrix} \)\( = -3 \)
Cofactors of elements in row 2
Cofactor of \(a_{21} = (-1)^{2+1}\cdot |minor(a_{21})|\) \( = (-1)^{3}\cdot \begin{vmatrix}7\end{vmatrix} \)\( = -7 \)
Cofactor of \(a_{22} = (-1)^{2+2}\cdot |minor(a_{22})|\) \( = (-1)^{4}\cdot \begin{vmatrix}5\end{vmatrix} \)\( = 5 \)
Consolidate all the cofactors computed for respective elements in the matrix A to get the Cofactor Matrix of A.
Result
Therefore, the cofactor matrix of given matrix A, is
Cofactor Matrix of A \( = \begin{bmatrix}1 & -3\\-7 & 5\end{bmatrix} \)