Find the cofactors and determinants of 3x3 matrix A (10 marks)
A= [
8 2 3
1 9 4
4 3 6
]
To find the cofactors and determinants of a 3x3 matrix A, we will first find the minor matrix by taking the determinant of each of the 2x2 matrices formed by removing the row and column of each element of the matrix.
Matrix A:
8 2 3
1 9 4
4 3 6
Minor matrix:
|9 4| |4 3| |3 6|
|3 6| |1 4| |8 3|
Calculating the minor matrix:
|9 4| = 9*6 - 4*3 = 54 - 12 = 42
|3 6|
|4 3| = 1*6 - 4*3 = 6 - 12 = -6
|8 3|
|3 6| = 1*3 - 9*6 = 3 - 54 = -51
|8 3|
Now we will find the cofactors by adding alternate signs to the elements of the minor matrix.
Cofactor matrix:
|42 -6 51|
| -6 42 -6|
|51 -6 42|
Finally, we will find the determinant of matrix A by using the formula:
det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
det(A) = 8(9*6 - 4*3) - 2(1*6 - 4*3) + 3(1*3 - 9*6)
det(A) = 8(54 - 12) - 2(6 - 12) + 3(3 - 54)
det(A) = 8(42) - 2(-6) + 3(-51)
det(A) = 336 + 12 - 153
det(A) = 195
Therefore, the cofactors of matrix A are given by:
|42 -6 51|
| -6 42 -6|
|51 -6 42|
And the determinant of matrix A is 195.