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3131313131351515151515≈83137
How to find the inverse using the adjugate matrix
Compute every cofactor of the matrix, form the cofactor matrix, transpose it to obtain the adjugate, and divide each entry by the determinant of the original matrix. The result is the matrix inverse, provided the determinant is non-zero.
Adjugate matrix inverse — worked example (3×3)
Write the initial matrix
A
:
A
=
4
0
1
1
3
2
2
1
5
To calculate the inverse matrix of matrix
A
need to do the following:
1)
Calculate the determinant of the matrix A, and check whether it is not zero:If the determinant of the matrix A is not equal to zero, then we can continue the solution;
If the determinant of the matrix A is zero, it's inverse matrix cannot be calculated, because the matrix A is singular;
2)
Calculate the matrix of minors;3)
Calculate the matrix of cofactors;4)
Calculate the adjoint matrix;5)
Calculate the inverse matrix by finding the product of each element of the adjoint matrix by 1/d;a
-1
i,j
=
adj
0
i,j
*
1
d
i
is the row numberj
is the column numbera⁻¹
is element of the inverse matrixadj
is element of the adjoint matrixd
is the determinant of the matrix A2
Determinantdet(
A
) =
4
0
1
1
3
2
2
1
5
=
0
;
3
Inverse matrixThe inverse matrix cannot be calculated, because the matrix is singular (its determinant is equal to zero).