Cofactor Matrix Calculator
Enter the elements of the square matrix (2x2 up to 6x6)
to calculate its cofactor matrix step by step.
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2x2 matrix (integers)
2x2 matrix (fractions)
3x3 matrix (basic)
3x3 matrix (fractions)
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4x4 matrix (basic)
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Solved Exercises
The following are examples of problems solved by the calculator.
Exercise 1
Find the cofactor matrix of the following 3x3 matrix:
$$ \displaystyle A = \begin{bmatrix} 2 & -1 & 3 \\[0.5em] 0 & 4 & -2 \\[0.5em] 1 & 5 & 1 \end{bmatrix} $$
Result
The cofactor matrix of A is:
$$ \displaystyle C(A) = \begin{bmatrix} 14 & -2 & -4 \\[0.5em] 16 & -1 & -11 \\[0.5em] -10 & 4 & 8 \end{bmatrix} $$
Step-by-step solution
1. Determine the order of the cofactor matrix and its formula.
The cofactor matrix of A has the same order as the original matrix, but each element is replaced by its corresponding cofactor.
$$ \displaystyle C(A) = \begin{bmatrix} C_{11} & C_{12} & C_{13} \\[0.5em] C_{21} & C_{22} & C_{23} \\[0.5em] C_{31} & C_{32} & C_{33} \end{bmatrix} $$
Each element Cij is calculated using the following formula, where Mij is called the minor: the determinant of the submatrix resulting from removing row i and column j of matrix A:
$$ \displaystyle C_{ij} = (-1)^{i+j} M_{ij} $$
2. Calculate each cofactor.
To see the step-by-step calculation of each determinant, check out the determinant calculator .
Calculation of C11 $$ \displaystyle C_{11} = (-1)^{1+1} M_{11} \\[0.8em] = (-1)^{2} \begin{vmatrix} 4 & -2 \\[0.5em] 5 & 1 \end{vmatrix} \\[0.8em] = (1) (14) \\[0.8em] = 14 $$
Calculation of C12 $$ \displaystyle C_{12} = (-1)^{1+2} M_{12} \\[0.8em] = (-1)^{3} \begin{vmatrix} 0 & -2 \\[0.5em] 1 & 1 \end{vmatrix} \\[0.8em] = (-1) (2) \\[0.8em] = -2 $$
Calculation of C13 $$ \displaystyle C_{13} = (-1)^{1+3} M_{13} \\[0.8em] = (-1)^{4} \begin{vmatrix} 0 & 4 \\[0.5em] 1 & 5 \end{vmatrix} \\[0.8em] = (1) (-4) \\[0.8em] = -4 $$
Calculation of C21 $$ \displaystyle C_{21} = (-1)^{2+1} M_{21} \\[0.8em] = (-1)^{3} \begin{vmatrix} -1 & 3 \\[0.5em] 5 & 1 \end{vmatrix} \\[0.8em] = (-1) (-16) \\[0.8em] = 16 $$
Calculation of C22 $$ \displaystyle C_{22} = (-1)^{2+2} M_{22} \\[0.8em] = (-1)^{4} \begin{vmatrix} 2 & 3 \\[0.5em] 1 & 1 \end{vmatrix} \\[0.8em] = (1) (-1) \\[0.8em] = -1 $$
Calculation of C23 $$ \displaystyle C_{23} = (-1)^{2+3} M_{23} \\[0.8em] = (-1)^{5} \begin{vmatrix} 2 & -1 \\[0.5em] 1 & 5 \end{vmatrix} \\[0.8em] = (-1) (11) \\[0.8em] = -11 $$
Calculation of C31 $$ \displaystyle C_{31} = (-1)^{3+1} M_{31} \\[0.8em] = (-1)^{4} \begin{vmatrix} -1 & 3 \\[0.5em] 4 & -2 \end{vmatrix} \\[0.8em] = (1) (-10) \\[0.8em] = -10 $$
Calculation of C32 $$ \displaystyle C_{32} = (-1)^{3+2} M_{32} \\[0.8em] = (-1)^{5} \begin{vmatrix} 2 & 3 \\[0.5em] 0 & -2 \end{vmatrix} \\[0.8em] = (-1) (-4) \\[0.8em] = 4 $$
Calculation of C33 $$ \displaystyle C_{33} = (-1)^{3+3} M_{33} \\[0.8em] = (-1)^{6} \begin{vmatrix} 2 & -1 \\[0.5em] 0 & 4 \end{vmatrix} \\[0.8em] = (1) (8) \\[0.8em] = 8 $$
3. Construct the final matrix.
Having the values of all cofactors, we determine that the cofactor matrix is:
$$ \displaystyle C(A) = \begin{bmatrix} 14 & -2 & -4 \\[0.5em] 16 & -1 & -11 \\[0.5em] -10 & 4 & 8 \end{bmatrix} $$
Exercise 2
Calculate the cofactor matrix given the matrix:
$$ \displaystyle A = \begin{bmatrix} \frac{1}{3} & -1 & 2 \\[0.5em] 0 & \frac{1}{4} & 3 \\[0.5em] -2 & 0 & 1 \end{bmatrix} $$
Result
We obtain the following cofactor matrix of A as a result:
$$ \displaystyle C(A) = \begin{bmatrix} \frac{1}{4} & -6 & \frac{1}{2} \\[0.5em] 1 & \frac{13}{3} & 2 \\[0.5em] -\frac{7}{2} & -1 & \frac{1}{12} \end{bmatrix} $$
Step-by-step solution
1. Determine the order of the cofactor matrix and its formula.
The cofactor matrix of A has the same order as the original matrix, but each element is replaced by its corresponding cofactor.
$$ \displaystyle C(A) = \begin{bmatrix} C_{11} & C_{12} & C_{13} \\[0.5em] C_{21} & C_{22} & C_{23} \\[0.5em] C_{31} & C_{32} & C_{33} \end{bmatrix} $$
Each element Cij is calculated using the following formula, where Mij is the determinant of the submatrix resulting from removing row i and column j of matrix A:
$$ \displaystyle C_{ij} = (-1)^{i+j} M_{ij} $$
2. Calculate each cofactor.
Calculation of C11 $$ \displaystyle C_{11} = (-1)^{1+1} M_{11} \\[0.8em] = (-1)^{2} \begin{vmatrix} \frac{1}{4} & 3 \\[0.5em] 0 & 1 \end{vmatrix} \\[0.8em] = (1) \left(\frac{1}{4}\right) \\[0.8em] = \frac{1}{4} = 0.25 $$
Calculation of C12 $$ \displaystyle C_{12} = (-1)^{1+2} M_{12} \\[0.8em] = (-1)^{3} \begin{vmatrix} 0 & 3 \\[0.5em] -2 & 1 \end{vmatrix} \\[0.8em] = (-1) (6) \\[0.8em] = -6 $$
Calculation of C13 $$ \displaystyle C_{13} = (-1)^{1+3} M_{13} \\[0.8em] = (-1)^{4} \begin{vmatrix} 0 & \frac{1}{4} \\[0.5em] -2 & 0 \end{vmatrix} \\[0.8em] = (1) \left(\frac{1}{2}\right) \\[0.8em] = \frac{1}{2} = 0.5 $$
Calculation of C21 $$ \displaystyle C_{21} = (-1)^{2+1} M_{21} \\[0.8em] = (-1)^{3} \begin{vmatrix} -1 & 2 \\[0.5em] 0 & 1 \end{vmatrix} \\[0.8em] = (-1) (-1) \\[0.8em] = 1 $$
Calculation of C22 $$ \displaystyle C_{22} = (-1)^{2+2} M_{22} \\[0.8em] = (-1)^{4} \begin{vmatrix} \frac{1}{3} & 2 \\[0.5em] -2 & 1 \end{vmatrix} \\[0.8em] = (1) \left(\frac{13}{3}\right) \\[0.8em] = \frac{13}{3} \approx 4.33 $$
Calculation of C23 $$ \displaystyle C_{23} = (-1)^{2+3} M_{23} \\[0.8em] = (-1)^{5} \begin{vmatrix} \frac{1}{3} & -1 \\[0.5em] -2 & 0 \end{vmatrix} \\[0.8em] = (-1) (-2) \\[0.8em] = 2 $$
Calculation of C31 $$ \displaystyle C_{31} = (-1)^{3+1} M_{31} \\[0.8em] = (-1)^{4} \begin{vmatrix} -1 & 2 \\[0.5em] \frac{1}{4} & 3 \end{vmatrix} \\[0.8em] = (1) \left(-\frac{7}{2}\right) \\[0.8em] = -\frac{7}{2} = -3.5 $$
Calculation of C32 $$ \displaystyle C_{32} = (-1)^{3+2} M_{32} \\[0.8em] = (-1)^{5} \begin{vmatrix} \frac{1}{3} & 2 \\[0.5em] 0 & 3 \end{vmatrix} \\[0.8em] = (-1) (1) \\[0.8em] = -1 $$
Calculation of C33 $$ \displaystyle C_{33} = (-1)^{3+3} M_{33} \\[0.8em] = (-1)^{6} \begin{vmatrix} \frac{1}{3} & -1 \\[0.5em] 0 & \frac{1}{4} \end{vmatrix} \\[0.8em] = (1) \left(\frac{1}{12}\right) \\[0.8em] = \frac{1}{12} \approx 0.08 $$
3. Construct the final matrix.
Having the values of all cofactors, we determine that the cofactor matrix is:
$$ \displaystyle C(A) = \begin{bmatrix} \frac{1}{4} & -6 & \frac{1}{2} \\[0.5em] 1 & \frac{13}{3} & 2 \\[0.5em] -\frac{7}{2} & -1 & \frac{1}{12} \end{bmatrix} $$
Exercise 3
Determine the cofactor matrix of the 2x2 matrix:
$$ \displaystyle A = \begin{bmatrix} 3 & 4 \\[0.5em] -2 & 5 \end{bmatrix} $$
Result
The corresponding cofactor matrix of A is:
$$ \displaystyle C(A) = \begin{bmatrix} 5 & 2 \\[0.5em] -4 & 3 \end{bmatrix} $$
Step-by-step solution
1. Determine the order of the cofactor matrix and its formula.
The cofactor matrix of A has the same order as the original matrix, but each element is replaced by its corresponding cofactor.
$$ \displaystyle C(A) = \begin{bmatrix} C_{11} & C_{12} \\[0.5em] C_{21} & C_{22} \end{bmatrix} $$
Each element Cij is calculated using the following formula, where Mij is the determinant of the submatrix resulting from removing row i and column j of matrix A:
$$ \displaystyle C_{ij} = (-1)^{i+j} M_{ij} $$
2. Calculate each cofactor.
Calculation of C11 $$ \displaystyle C_{11} = (-1)^{1+1} M_{11} \\[0.8em] = (-1)^{2} \begin{vmatrix} 5 \end{vmatrix} \\[0.8em] = (1) (5) \\[0.8em] = 5 $$
Calculation of C12 $$ \displaystyle C_{12} = (-1)^{1+2} M_{12} \\[0.8em] = (-1)^{3} \begin{vmatrix} -2 \end{vmatrix} \\[0.8em] = (-1) (-2) \\[0.8em] = 2 $$
Calculation of C21 $$ \displaystyle C_{21} = (-1)^{2+1} M_{21} \\[0.8em] = (-1)^{3} \begin{vmatrix} 4 \end{vmatrix} \\[0.8em] = (-1) (4) \\[0.8em] = -4 $$
Calculation of C22 $$ \displaystyle C_{22} = (-1)^{2+2} M_{22} \\[0.8em] = (-1)^{4} \begin{vmatrix} 3 \end{vmatrix} \\[0.8em] = (1) (3) \\[0.8em] = 3 $$
3. Construct the final matrix.
Having the values of all cofactors, we determine that the cofactor matrix is:
$$ \displaystyle C(A) = \begin{bmatrix} 5 & 2 \\[0.5em] -4 & 3 \end{bmatrix} $$
Exercise 4
Find the cofactor matrix of the 4x4 matrix:
$$ \displaystyle A = \begin{bmatrix} 1 & 0 & 2 & -1 \\[0.5em] 3 & 1 & 0 & 2 \\[0.5em] -2 & 0 & 1 & 1 \\[0.5em] 4 & -1 & 3 & 0 \end{bmatrix} $$
Result
The resulting cofactor matrix for A is:
$$ \displaystyle C(A) = \begin{bmatrix} -1 & 29 & 11 & -13 \\[0.5em] 3 & 15 & 1 & 5 \\[0.5em] -7 & -1 & 9 & 11 \\[0.5em] 3 & -19 & 1 & 5 \end{bmatrix} $$
Step-by-step solution
1. Determine the order of the cofactor matrix and its formula.
The cofactor matrix of A has the same order as the original matrix, but each element is replaced by its corresponding cofactor.
$$ \displaystyle C(A) = \begin{bmatrix} C_{11} & C_{12} & C_{13} & C_{14} \\[0.5em] C_{21} & C_{22} & C_{23} & C_{24} \\[0.5em] C_{31} & C_{32} & C_{33} & C_{34} \\[0.5em] C_{41} & C_{42} & C_{43} & C_{44} \end{bmatrix} $$
Each element Cij is calculated using the following formula, where Mij is the determinant of the submatrix resulting from removing row i and column j of matrix A:
$$ \displaystyle C_{ij} = (-1)^{i+j} M_{ij} $$
2. Calculate each cofactor.
Calculation of C11 $$ \displaystyle C_{11} = (-1)^{1+1} M_{11} \\[0.8em] = (-1)^{2} \begin{vmatrix} 1 & 0 & 2 \\[0.5em] 0 & 1 & 1 \\[0.5em] -1 & 3 & 0 \end{vmatrix} \\[0.8em] = (1) (-1) \\[0.8em] = -1 $$
Calculation of C12 $$ \displaystyle C_{12} = (-1)^{1+2} M_{12} \\[0.8em] = (-1)^{3} \begin{vmatrix} 3 & 0 & 2 \\[0.5em] -2 & 1 & 1 \\[0.5em] 4 & 3 & 0 \end{vmatrix} \\[0.8em] = (-1) (-29) \\[0.8em] = 29 $$
Calculation of C13 $$ \displaystyle C_{13} = (-1)^{1+3} M_{13} \\[0.8em] = (-1)^{4} \begin{vmatrix} 3 & 1 & 2 \\[0.5em] -2 & 0 & 1 \\[0.5em] 4 & -1 & 0 \end{vmatrix} \\[0.8em] = (1) (11) \\[0.8em] = 11 $$
Calculation of C14 $$ \displaystyle C_{14} = (-1)^{1+4} M_{14} \\[0.8em] = (-1)^{5} \begin{vmatrix} 3 & 1 & 0 \\[0.5em] -2 & 0 & 1 \\[0.5em] 4 & -1 & 3 \end{vmatrix} \\[0.8em] = (-1) (13) \\[0.8em] = -13 $$
Calculation of C21 $$ \displaystyle C_{21} = (-1)^{2+1} M_{21} \\[0.8em] = (-1)^{3} \begin{vmatrix} 0 & 2 & -1 \\[0.5em] 0 & 1 & 1 \\[0.5em] -1 & 3 & 0 \end{vmatrix} \\[0.8em] = (-1) (-3) \\[0.8em] = 3 $$
Calculation of C22 $$ \displaystyle C_{22} = (-1)^{2+2} M_{22} \\[0.8em] = (-1)^{4} \begin{vmatrix} 1 & 2 & -1 \\[0.5em] -2 & 1 & 1 \\[0.5em] 4 & 3 & 0 \end{vmatrix} \\[0.8em] = (1) (15) \\[0.8em] = 15 $$
Calculation of C23 $$ \displaystyle C_{23} = (-1)^{2+3} M_{23} \\[0.8em] = (-1)^{5} \begin{vmatrix} 1 & 0 & -1 \\[0.5em] -2 & 0 & 1 \\[0.5em] 4 & -1 & 0 \end{vmatrix} \\[0.8em] = (-1) (-1) \\[0.8em] = 1 $$
Calculation of C24 $$ \displaystyle C_{24} = (-1)^{2+4} M_{24} \\[0.8em] = (-1)^{6} \begin{vmatrix} 1 & 0 & 2 \\[0.5em] -2 & 0 & 1 \\[0.5em] 4 & -1 & 3 \end{vmatrix} \\[0.8em] = (1) (5) \\[0.8em] = 5 $$
Calculation of C31 $$ \displaystyle C_{31} = (-1)^{3+1} M_{31} \\[0.8em] = (-1)^{4} \begin{vmatrix} 0 & 2 & -1 \\[0.5em] 1 & 0 & 2 \\[0.5em] -1 & 3 & 0 \end{vmatrix} \\[0.8em] = (1) (-7) \\[0.8em] = -7 $$
Calculation of C32 $$ \displaystyle C_{32} = (-1)^{3+2} M_{32} \\[0.8em] = (-1)^{5} \begin{vmatrix} 1 & 2 & -1 \\[0.5em] 3 & 0 & 2 \\[0.5em] 4 & 3 & 0 \end{vmatrix} \\[0.8em] = (-1) (1) \\[0.8em] = -1 $$
Calculation of C33 $$ \displaystyle C_{33} = (-1)^{3+3} M_{33} \\[0.8em] = (-1)^{6} \begin{vmatrix} 1 & 0 & -1 \\[0.5em] 3 & 1 & 2 \\[0.5em] 4 & -1 & 0 \end{vmatrix} \\[0.8em] = (1) (9) \\[0.8em] = 9 $$
Calculation of C34 $$ \displaystyle C_{34} = (-1)^{3+4} M_{34} \\[0.8em] = (-1)^{7} \begin{vmatrix} 1 & 0 & 2 \\[0.5em] 3 & 1 & 0 \\[0.5em] 4 & -1 & 3 \end{vmatrix} \\[0.8em] = (-1) (-11) \\[0.8em] = 11 $$
Calculation of C41 $$ \displaystyle C_{41} = (-1)^{4+1} M_{41} \\[0.8em] = (-1)^{5} \begin{vmatrix} 0 & 2 & -1 \\[0.5em] 1 & 0 & 2 \\[0.5em] 0 & 1 & 1 \end{vmatrix} \\[0.8em] = (-1) (-3) \\[0.8em] = 3 $$
Calculation of C42 $$ \displaystyle C_{42} = (-1)^{4+2} M_{42} \\[0.8em] = (-1)^{6} \begin{vmatrix} 1 & 2 & -1 \\[0.5em] 3 & 0 & 2 \\[0.5em] -2 & 1 & 1 \end{vmatrix} \\[0.8em] = (1) (-19) \\[0.8em] = -19 $$
Calculation of C43 $$ \displaystyle C_{43} = (-1)^{4+3} M_{43} \\[0.8em] = (-1)^{7} \begin{vmatrix} 1 & 0 & -1 \\[0.5em] 3 & 1 & 2 \\[0.5em] -2 & 0 & 1 \end{vmatrix} \\[0.8em] = (-1) (-1) \\[0.8em] = 1 $$
Calculation of C44 $$ \displaystyle C_{44} = (-1)^{4+4} M_{44} \\[0.8em] = (-1)^{8} \begin{vmatrix} 1 & 0 & 2 \\[0.5em] 3 & 1 & 0 \\[0.5em] -2 & 0 & 1 \end{vmatrix} \\[0.8em] = (1) (5) \\[0.8em] = 5 $$
3. Construct the final matrix.
Having the values of all cofactors, we determine that the cofactor matrix is:
$$ \displaystyle C(A) = \begin{bmatrix} -1 & 29 & 11 & -13 \\[0.5em] 3 & 15 & 1 & 5 \\[0.5em] -7 & -1 & 9 & 11 \\[0.5em] 3 & -19 & 1 & 5 \end{bmatrix} $$
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