Find the cofactors and determinats of 3×3 matrix A
[8 1 3]
A= [1 9 4]
[4 3 6]
To find the cofactors of a 3x3 matrix, we first need to find the minors of each element. The minor of element a_ij is the determinant of the 2x2 matrix formed by removing the i-th row and j-th column of the original matrix.
For the matrix A:
minors:
M_11 = |9 4|
|3 6| = (9*6 - 4*3) = 54 - 12 = 42
M_12 = |1 4|
|4 6| = (1*6 - 4*4) = 6 - 16 = -10
M_13 = |1 9|
|4 3| = (1*3 - 9*4) = 3 - 36 = -33
M_21 = |1 4|
|3 6| = (1*6 - 4*3) = 6 - 12 = -6
M_22 = |8 3|
|4 6| = (8*6 - 3*4) = 48 - 12 = 36
M_23 = |8 1|
|4 3| = (8*3 - 1*4) = 24 - 4 = 20
M_31 = |1 4|
|9 4| = (1*4 - 4*9) = 4 - 36 = -32
M_32 = |8 3|
|1 6| = (8*6 - 3*1) = 48 - 3 = 45
M_33 = |8 1|
|1 9| = (8*9 - 1*1) = 72 - 1 = 71
Now, we can find the cofactors by multiplying each minor by (-1)^(i+j), where i and j are the row and column indices of the element:
Cofactors:
C_11 = M_11 = 42
C_12 = -M_12 = 10
C_13 = M_13 = -33
C_21 = -M_21 = 6
C_22 = M_22 = 36
C_23 = -M_23 = -20
C_31 = M_31 = -32
C_32 = -M_32 = -45
C_33 = M_33 = 71
Therefore, the cofactors of matrix A are:
[42 10 -33]
[6 36 -20]
[-32 -45 71]
Finally, to find the determinant of matrix A, we can sum the product of each element of a row or column with its corresponding cofactor:
det(A) = 8*C_11 + 1*C_12 + 3*C_13
= 8*42 + 1*10 + 3*(-33)
= 336 + 10 - 99
= 247
Therefore, the determinant of matrix A is 247.