Maximus Bond meets Jill Bonza (who is related to our coordinator and security expert for this site Jim Bonza)
somewhere deep inside Linear Spy headquarters.
Hello Max, what are you doing in Matrix World?
Hello Jill. Matrix World? I didn't know I was in Matrix World. This must be where they made the movie "Matrix"! Cool!
No Max. This place is all about the mathematics of matrices. Its nothing to do with the movie.
Matrices? I thought you said matrix.
We say one matrix, but the plural is matrices, Max.
Well, I like mathematics Jill, so, now that I'm here, why don't you tell me a bit about matrices?
Well Max , they are very useful in all sorts of areas, so it would be worth spending a bit of time finding out about them. Jim Bonza likes to use them for secret codes. Start here Max, here is the definition.
A matrix is a rectangular array of items. In our context the items will typically be numbers.
EXAMPLES. A and B below are examples of
Most books write the array inside large brackets, but that is difficult here, so we will use a border right around instead.
TERMINOLOGY AND NOTATION
- If a matrix has m rows and n columns we say it is an mxn matrix or that the SIZE of the matrix is mxn.
In the example above, A is a 3x2 matrix.
- When m = n we say the matrix is SQUARE. In the example above, B is a square matrix and has size 2x2.
- The numbers in the array are called ELEMENTS or ENTRIES of the matrix.
- If we want to write a general matrix A, we denote the element in row i and column j as aij.
So a shorthand way of writing A is A = (aij) where i = 1, 2, ...m and j = 1, 2, ...n and by this we
| A = ||
| a11 || a12 || ... ||
| a21 || a22 || ... ||
| ... || ... || ... || ... |
| am1 || am2 || ... ||
For A in the example above, a22 = 0, a31 = -2. For B in the example above, b12 = 9, b22 =
- If a matrix has only one row it is called a ROW MATRIX or
ROW VECTOR; If a matrix has only one
column it is called a COLUMN MATRIX or
EXAMPLES. C below is an example of a row matrix; D below is a column matrix.
BASIC PROPERTIES AND OPERATIONS
Two matrices are equalif and only if they have the same size and corresponding elements are equal.
EXAMPLE. When does
The two matrices have the same size. To be equal we also need
| a = 1, b = 2, c = 3, d = 4.|
A zero matrix is a matrix with all its entries zero. We denote the zero matrix by 0 if there is no
confusion about size, otherwise we use 0mxn.
If A and B are the same size we can add (subtract) them and we do so by adding (subtracting)
is not defined because the matrices are different sizes.
PROPERTIES OF THE ADDITION OPERATION
If A, B, C are all mxn matrices (i.e. all the same size) then
A + B = B + A
A + (B + C) = (A + B) + C
A + 0mxn = A.
If A is any matrix and p is any real number, the product pA is the matrix obtained by
multiplying each element of A by p.
PROPERTIES OF THE SCALAR MULTIPLICATION OPERATION
If p and q are real numbers and A and B are matrices of the same size, then
p(A + B) = pA + pB
(p + q)A = pA + qA
pqA = p(qA)
TRANSPOSE OF A MATRIX
If A is an mxn matrix then the transpose of A, denoted AT or At or
sometimes A', is the nxm matrix obtained by interchanging the rows and
columns of A.
Thus the first row becomes the first column, the second row becomes the second column, etc.
EXAMPLE. Find the transpose of
The transpose is
PROPERTIES OF THE TRANSPOSE OPERATION
If A and B are matrices of the same size, then
(AT)T = A
(A + B)T = AT + BT.
Max is looking very excited:
Jill, Jill, I thought you said this place had nothing to do
with the movie 'Matrix", but isn't that Trinity over there?
Yes max that is Trinity. She helps us out sometimes. She is over in the
Is she getting an undercover identity Jill?
No Max. Even matrices have identities you know.
Why don't you go on to the tutOR page on Matrix
Product to see what we mean by AB, i.e. two matrices multiplied together, and to learn about matrix
I think I will, will you join me there, readers?