Lecture Notes in Differential Equations - Bruce E. Shapiro

121
Definition 14.19. Matrix Multiplication. Let A = [a ij ] be an m × r
matrix and let B = [b ij ] be an r × n matrix. Then the matrix product is
defined by
r∑
[AB] ij = a ik b kr = row i (A) · column j B (14.24)
k=1
i.e., the ij th element of the product is the dot product between the i th row
of A and the j th column of B.
Example 14.1.
( ) ⎛ ⎞
8 9 ( )
1 2 3
⎝10 11⎠ (1, 2, 3) · (8, 10, 12) (1, 2, 3) · (9, 11, 13)
=
4 5 6
(4, 5, 6) · (8, 10, 12) (4, 5, 6) · (9, 11, 13)
12 13
(14.25)
( ) 64 70
=
(14.26)
156 169
Note that the product of an [n × r] matrix and an [r × m] matrix is always
an [n × m] matrix. The product of an [n × r] matrix and and [s × n] is
undefined unless r = s.
Theorem 14.20. If A and B are both n × n square matrices then
det AB = (det A)(det B) (14.27)
Definition 14.21. Identity Matrix. The n × n matrix I is defined as
the matrix with 1’s in the main diagonal a 11 , a 22 , . . . , a mm and zeroes
everywhere else.
Theorem 14.22. I is the identity under matrix multiplication. Let A be
any n × n matrix and I the n × n Identity matrix. Then AI = IA = A.
Definition 14.23. A square matrix A is said to be invertible if there
exists a matrix A −1 , called the inverse of A, such that
AA −1 = A −1 A = I (14.28)
Theorem 14.24. A square matrix is invertible if and only if it is nonsingular,
i.,e, det A ≠ 0.
Definition 14.25. Let A = [a ij ] be any square matrix of order n. Then
the cofactor of a ij , denoted by cof a ij , is the (−1) i+j det M ij where M ij
is the submatrix of A with row i and column j removed.