The adjoint of the zero matrix (or null matrix) results in the zero matrix:.The adjoint of a matrix has the following characteristics: ➤ Did you know that the adjugate matrix is used to calculate the inverse of a matrix? See how to find the matrix inverse. There is no formula to directly find the adjoint of a 3×3 matrix. Secondly, we replace each element of matrix B by its cofactor to determine the cofactor matrix of B:Īnd finally, we transpose the cofactor matrix to find the adjoint of matrix B: So, first we find the cofactor of each element of the matrix: To compute the adjoint of the 3×3 matrix we have to apply the same procedure. Let B be the following square matrix of order 3: You can verify the formula by calculating with it the example seen above. However, this formula is only valid for 2×2 matrices. On the other hand, there is a formula to find the adjoint of a 2×2 matrix without doing any calculations: Now we replace each element of matrix A by its cofactor to find the cofactor matrix of A:Īnd finally, we simply have to transpose the cofactor matrix: To do this, we have to apply the following formula: To compute the adjoint of matrix A, we first have to find the cofactor of each entry of the matrix.

Let A be the following square matrix of order 2: We will first see the adjoint of a 2×2 dimension matrix, and then the adjoint of a 3×3 dimension matrix. Having seen the theory of the adjoint of a matrix, here are some solved examples of the calculation of the adjoint of a matrix.

Note that the adjoint of a matrix can only be found for square matrices. Where M ij is the i, j minor of the matrix, that is, the determinant that results from deleting the i-th row and the j-th column of the matrix. Remember that the formula to compute the i, j cofactor of a matrix is as follows: To find the adjoint of a matrix, first replace each element in the matrix by its cofactor and then transpose the matrix.

Matrix adjoint formula