Adjoint Matrix Calculator

The adjoint (often called the adjugate) of a square matrix A is defined as the transpose of its cofactor matrix, denoted as adj(A) = C^T. To find the adjoint, you must first construct the matrix of cofactors. For any element a_ij, its cofactor C_ij is calculated by taking the determinant of the submatrix formed by removing the i-th row and j-th column, multiplied by the sign factor (-1)^(i+j). Once all cofactors are computed, transposing the resulting matrix by swapping its rows and columns yields the final adjoint matrix.

Let us find the adjoint of a 2x2 matrix A defined as:

| 2   1 |
| 4   3 |

Step 1: Find the cofactor for each element.
For a 2x2 matrix, the cofactor matrix elements are found by swapping diagonal elements and changing the signs of the off-diagonals.
C_11 = 3
C_12 = -4
C_21 = -1
C_22 = 2

Step 2: Construct the cofactor matrix C.
| 3   -4 |
| -1   2 |

Step 3: Transpose the cofactor matrix to get adj(A).
adj(A) = C^T =
| 3   -1 |
| -4   2 |

When working with matrix calculations, always verify that your matrix is square (having an equal number of rows and columns) because non-square matrices do not possess an adjoint. Pay close attention to alternating signs (-1)^(i+j) during cofactor expansion, as a single sign error will corrupt the entire output. Finally, use the relationship A * adj(A) = det(A) * I to check your work and confirm that the resulting matrix is correct.