A system when b1 = b2 = ··· = bm = 0 is called
Homogeneous
Non-homogeneous
A system is said to be homogeneous when b1 = b2 = ·· ·= bm = 0
Homogeneous systems always have at least one solution called “the trivial solution”
Homogeneous systems don't have solutions
Number of rows
m
n
Number of columns
( 5 6 7 8 ) is a 1 x 4 matrix
( 5 6 7 8 ) is a 4 x 1 matrix
( 5 6 7 8 9 ) is a 5 x 1 matrix
( 5 6 7 8 9 ) is a 1 x 5 matrix
The matrix 1 2 3 4 Is a 1 x 4 matrix
The matrix 1 2 3 4 Is a 4 x 1 matrix
What is the order of The matrix 1 2 3 4
1 x 4
4 x 1
4
1
Two matrices A and B are equal if and only if:
Both have the same dimension say m x n aij = bij for all i,j.
Both have the same dimension say m x n a = bij for all i,j.
Both have the same dimension say m x n aij = bij for some i,j.
Both have the same dimension say m x n aij = b for all i,j.
A + (-A) =
0mxn
0
A
A-A
Matrix Multiplication Is Not Commutative
Matrix Multiplication Is Commutative
Matrix Multiplication Is Not Associative
Matrix Multiplication Is Associative
What does it mean for matrix multiplication to be associative ?
ABC = A(BC) = (AB)C
ABC = A(BC) = ABC
ABC = CBA
ABC = CBA = CAB = BCA
An n × n matrix is diagonal when: aij =0 for i /= j
An n × n matrix is diagonal when: aij =0 for i = j
An upper triangular matrix is any n × n matrix where
aij =0 where i /= j
aij =0 wherei > j
aij =0 where i > i
lower triangular matrix has:
aij =0 where j > i
aij =0 where j \> i
aij =0 where j = i
If the n x n matrix A is invertible then
AB = In = BA
AB = In = BA-1
AB = BA
AB = In = B-1A-1
