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A matrix is just a rectangular array ("grid") of numbers. Here are a few.
||,||[2 -10 -1]||,||
The above matrix has 4 rows and 3 columns. We refer to it as a 43 matrix. Here is the general definition from Finite Mathematics.
Matrix Addition and Subtraction
Two matrices can be added (or subtracted) if, and only if, they have the same dimensions. (That is, both matrices have matching numbers of rows and columns. For instance, you can't add, say, a 34 matrix to a 44 matrix, but you can add two 34 matrices.)
To add (or subtract) two matrices of the same dimensions, just add (or subtract) the corresponding entries. In other words, if A and B are mn matrices, then A+B and A-B are the mn matrices whose entries are given by
|(A + B)ij||=||Aij + Bij||ijth entry of the sum = sum of the ijth entries|
|(A - B)ij||=||Aij - Bij||ijth entry of the difference = difference of the ijth entries|
A matrix all of whose entries are zero is called a zero matrix. Here are some zero matrices.
||,||[0 0 0]||,||
It is traditional when talking about matrices to call individual numbers scalars. For this reason we call the operation of multiplying a matrix by a number scalar multiplication.
In order to motivate scalar multiplication, consider the following.
Q When can a matrix A be added to itself?
A Always, since the expression A+A is the sum of two matrices that certainly have the same dimensions.
Q Can't we write A+A as 2A?
A We certainly can. Notice that, when we compute A+A, we end up doubling every entry in A. Thus, we can think of the expression 2A as telling us to multiply every element in A by 2.
Similarly, 6A is the matrix obtained from A by multiplying each of its entries by 6. More generally, if c is any number, then cA ("c times A") is the matrix obtained from A by multiplying each of its entries by c.
Algebra of Matrices
Addition and scalar multiplcation of matrices satsify rules similar to those for addition and multiplication of real numbers.
Q What about the product of two matrices?
A That must wait till the next section (press "Next Tutorial" on the side if you're impatient to see matrix multiplication.)
Q OK the above rules are all very nice, but how can we apply them?
A Matrix algebra can be applied in a mind-boggling number of situations. Here is just one.
Now try some of the exercises in Section 3.1 of Finite Math, or Finite Math and Applied Calc.