Find the cofactors and determinants of 3x3 matrix A
A= [
8 2 3
1 9 4
4 3 6
]
To find the cofactors of a 3x3 matrix, we must first find the cofactor matrix. The cofactor of an element A[i,j] is calculated as follows:
C[i,j] = (-1)^(i+j) * M[i,j]
Where M[i,j] is the determinant of the 2x2 submatrix that remains after removing the row and column of A[i,j].
For matrix A:
C[1,1] = |9 4|
|3 6|
= (9*6) - (4*3)
= 54 - 12
= 42
C[1,2] = |1 4|
|4 6|
= (1*6) - (4*4)
= 6 - 16
= -10
C[1,3] = |1 9|
|4 3|
= (1*3) - (9*4)
= 3 - 36
= -33
C[2,1] = |2 3|
|3 6|
= (2*6) - (3*3)
= 12 - 9
= 3
C[2,2] = |8 3|
|4 6|
= (8*6) - (3*4)
= 48 - 12
= 36
C[2,3] = |8 2|
|4 3|
= (8*3) - (2*4)
= 24 - 8
= 16
C[3,1] = |2 3|
|9 4|
= (2*4) - (3*9)
= 8 - 27
= -19
C[3,2] = |8 3|
|1 4|
= (8*4) - (3*1)
= 32 - 3
= 29
C[3,3] = |8 2|
|1 9|
= (8*9) - (2*1)
= 72 - 2
= 70
Therefore, the cofactor matrix of A is:
C = [
42 -10 -33
3 36 16
-19 29 70
]
To find the determinant of matrix A, we can use any row or column to calculate it. Let's use the first row:
det(A) = 8*C[1,1] - 2*C[1,2] + 3*C[1,3]
= 8*42 - 2*(-10) + 3*(-33)
= 336 + 20 - 99
= 257
Therefore, the determinant of matrix A is 257.