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References
Robert Wrede and Murrary R. Spiegel, Theory
and Problems of Advanced Calculas, 2nd Edition,
2002.
Matrix Multiplication:
• We now need to define the concept of the product of two
matrices. Not All Matrices Can Be Multiplied: they must
have the right shape, or be conformable for multiplication
to be defined. The product of A and B, in this order, is
written as AB (no product sign is used), but it is only
defined if the number of columns in A equals the number
of rows in B. The product BA might not exist, and if it does,
it will not in general be equal to AB.
Let us look at the case where A is a 1x3 matrix, which is a row
vector, and B is a 3x1 matrix, which is a column vector, given
by:
The product AB is defined as the 1x1 matrix C given by:
Here, the single remaining element is the sum of the products of
corresponding elements from the row in A and the column in B,
Thus the product of a 1x3 matrix and a 3x1 matrix is a 1x1 matrix,
This is known as a row-on-column operation.
1x3 3x1
1x1
Suppose now that A is a 2x3 matrix and that B is a 3x2 matrix
which are given by:
The product AB is now a 2x2 matrix C given by:
Note that each row in A 'operates' on each column in B giving four elements in the 2x2 matrix C.
2x3 3x2
2x3 3x2
2x2
• Multiplication Rule: The element in the ith row and jth column of the product consist of the row-on-column product of the ith row A and jth column in B.
Example:If A is a 5x4 matrix, B is a 4x5 matrix and C is a 6x4 matrix, which is following products are defined: AB, BA, AC, CB, (AB)C, (CB)A?
AB is a 5x5 matrix BA is a 4x4 matrix AC is not defined
CB is a 6x5 matrix AB is a 5x5 matrix;
(AB) C is not defined
CB is a 6x5 matrix;
(CB) A is a 6x4 matrix
Example 1: Consider the matrices
Since A is a 2 × 3 matrix and B is a 3 × 4 matrix, the product
AB is a 2 × 4 matrix.
To determine, for example, the entry in row2 and column 3 of
AB , we single out row 2 from A and column 3 from B.
(2 · 4( + )6 · 3( + )0 · 5 = )26
The entry in row 1 and column 4 of AB is computed as follows:
(1 · 3( + )2 · 1( + )4 · 2 = )13
Exercise:
If:
Find AB and BA
Note: this example illustrates the point that AB be a zero
matrix without either A or B or AB being Zero.
A ( B + C ) = AB + AC (distributive law of addition)
A (BC) = (AB) C (associative law of multiplication)
• Special Matrices:
1. Transpose Matrix:
If A is any m × n matrix, then the transpose of A,
denoted by A, is defined to be the n × m matrix that
results from interchanging the rows and columns of A; that
is, the first column of A is the first row of A, the second
column of A is the second row of A, and so forth.
Example: The following are some examples of matrices and
their transposes.
Example:
Find AT, BT, A + BT and AB where:And confirm that (AB)T = BT AT
We see that:
Note That: (AB)T = BT AT
( A + B)T = AT + BT
2x3
3x2
2x3 3x2
2. Symmetric Matrices:
The square matrix is said to be symmetric if:
- A = AT Since rows and columns are interchanged in the
transpose, this is equivalent to aij = aji for elements if A =
[aij ]. Thus, Is a 3x3 symmetric matrix
- A = - AT Is a skew- symmetric matrix
3. Row and Column Vectors:
A row vector is a matrix one row, and a column vector is one
column. The transpose of a row vector is a column vector
and vice versa.
4. Diagonal Matrix:
A square matrix all of whose elements off the leading diagonal
Zero is called a diagonal matrix.
5. Identify Matrix:
The diagonal matrix with all diagonal elements 1 called the
identify or unit matrix. (AI =A , IA = A).
6. Power of Matrices:
If A is a square matrix of order nxn, then we write AA as A2, AA2
as A3 and so on.
If A is diagonal, as in:
Aⁿ = AA∙ ∙ ∙A )n >0(
n factors
Application For Multiplication:
If
Find the set of equations for x, y, z represented by Ax = d.
The set of linear equations for x, y, z is: x – y + 2z = 23x + y – 4z = 1 -x + 2y + z = -1