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| Basic Definitions
An m×n matrix A is a rectangular array of real numbers with m rows and n columns. (Rows are horizontal and columns are vertical.) The numbers m and n are the dimensions of A.
The real numbers in the matrix are called its entries. The entry in row i and column j is called aij or Aij.
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Example
Following is a 4×5 matrix with the entry A23 highlighted.
| A = |
 |
0 |
1 |
2 |
0 |
3 |
 |
|
1/3 |
-1 |
10 |
1/3 |
2 |
|
3 |
1 |
0 |
1 |
-3 |
|
2 |
1 |
0 |
0 |
1 |
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| Operations with Matrices
Transpose
The transpose, AT, of a matrix A is the matrix obtained from A by writing its rows as columns. If A is an m×n matrix and B = AT, then B is the n×m matrix with bij = aji.
Sum, Difference
If A and B have the same dimensions, then their sum, A+B, is obtained by adding corresponding entries. In symbols, (A+B)ij = Aij + Bij. If A and B have the same dimensions, then their difference, A - B, is obtained by subtracting corresponding entries. In symbols, (A-B)ij = Aij - Bij.
Scalar Multiple
If A is a matrix and c is a number (sometimes called a scalar in this context), then the scalar multiple, cA, is obtained by multiplying every entry in A by c. In symbols, (cA)ij = c(Aij).
Product
If A has dimensions m×n and B has dimensions n×p, then the product AB is defined, and has dimensions m×p. The entry (AB)ij is obtained by multiplying row i of A by column j of B, which is done by multiplying corresponding entries together and then adding the results. |
Examples
Transpose
|
 |
0 |
1 |
2 |
 |
T |
|
1/3 |
-1 |
10 | |
= |
 |
0 |
1/3 |
 |
|
1 |
-1 |
|
2 |
10 | |
Sum & Scalar Multiple
|
0 |
1 |
|
|
1/3 |
-1 | |
+ |
2 |
|
1 |
-1 |
|
|
2/3 |
-2 | |
= |
|
2 |
-1 |
|
|
5/3 |
-5 | | |