Matrix Multiplication

1
MatrixMultiplication
2
Cofactor
of the (i,j)th entry
(-1)ij (determinant obtained by deleting the ith
row and jth column of the matrix)
Aij
A11
A12
A13
next
3
Determinant
a21a32a13
a31a12a23
a11a22a33
direct expansion
a11a32a23
a31a22a13
a21a12a33
expandedalong
a11 A11 a12 A12 a13A13
1st row
a21 A21 a22 A22 a23A23
2nd row
4
Inverse
Proof
Explanation
Explanation
I.
Similarly, it can be proved that
A-1 hence found.
END
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Proof
back
6
Proof2
a11 A11 a12 A12 a13A13 det A
Expand along first row
0
a21 A21 a22 A22 a23A23 det A
Expand along second row
a31 A31 a32 A32 a33A33 det A
Expand along third row
back to inverse
a21 A11 a22 A12 a23A13?