The adjoint of a square matrix is another matrix defined in terms of the cofactors of the original matrix. If the inverse of the original matrix exists, the adjoint is "almost" that inverse, and can be used to find the inverse.

The adjoint of a square matrix A, denoted by
adj(A) is the matrix formed by

• replacing every entry of A by its cofactor
• taking the transpose of the result.

Here's the property that makes adjoints so useful. For any square matrix A, Aadj(A) = det(A)I.

Proof: The entry in row i and column j of Aadj(A) is the product of row i of A and column j of adj(A). The entries of column j of adj(A) are the cofactors of row j of A, so (because of the transpose) the entry in row i and column j of the product Aadj(A) is the product of row i of A and the column of cofactors from row j of A.

If i = j, we have the product of row i of A and the column of cofactors of row i, which is just the expansion of det(A) along row i.

If i ≠ j, we have the product of row i of A and a column of cofactors from a different row of A. This is the same as expanding a determinant with two equal rows (row i is repeated inside the cofactors), and so is 0.

The complete product then has det(A) along the main diagonal and zeroes elsewhere, and so is det(A)I.

Here's an inverse matrix found using this formula.

(The T for transpose is big and red as a reminder, in case you got absorbed in finding the cofactors and forgot to take the transpose at the end.)

Notice that det(A) is a scalar, so on the right-hand side of the equation Aadj(A) = det(A)I, we have a scalar multiple of the identity matrix. If det(A) is not zero, you can divide by that scalar and arrange the left-hand side appropriately to get

.

This equation says that the matrix inside the brackets is the inverse of A , i.e.

.

To summarize: to find the inverse of a matrix using cofactors

1. Find the determinant of the matrix, and make sure it's not zero (else the inverse doesn't exist).
2. Find the adjoint of the matrix
3. Divide the adjoint by the determinant to get the inverse.