To find the inverse of a 3x3 matrix, you can use the following formula:
Let's say you have a matrix A:
A = [a b c
d e f
g h i]
- Calculate the determinant of matrix A. In this case, the determinant of A is: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
- If the determinant of A is non-zero (det(A) ≠ 0), then the matrix A is invertible.
- Calculate the matrix of minors by replacing each element in A with the determinant of the 2x2 matrix formed by the remaining elements.
- For the element a, the minor is the determinant of the matrix [e f h i] [h i f g]
- For the element b, the minor is the determinant of the matrix [d f g i] [g i c e]
- For the element c, the minor is the determinant of the matrix [d e g h][g h b d]
- For the element d, the minor is the determinant of the matrix [b c h i] [h i a c]
- For the element e, the minor is the determinant of the matrix [a c g i] [g i f a]
- For the element f, the minor is the determinant of the matrix [a c d g] [g i b d]
- For the element g, the minor is the determinant of the matrix [b c f h] [h i a e]
- For the element h, the minor is the determinant of the matrix [a e g i] [g i d e]
- For the element, i, the minor is the determinant of the matrix [a e d g] [g i b e]
4. Create the matrix of cofactors by multiplying each element in the matrix of minors by (-1) raised to the power of the sum of its row and column numbers.
- For example, the cofactor of element a is (-1)^(1+1) times the minor of a.
5. Transpose the matrix of cofactors by swapping the elements along the main diagonal.
- For example, the element in the first row and second column becomes the element in the second row and first column.
6. Multiply the transposed matrix of cofactors by 1/det(A) to obtain the inverse of matrix A.
Applying these steps to the given matrix [1 0 0; 1 1 0; 1 1 1], you can calculate the inverse of the matrix.