Chapter 3 Matrices - Trinity College Dublin
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Chapter 3. Matrices
This material is in Chapter 1 of Anton & Rorres.
3.1 Basic matrix notation
We recall that amatrixis a rectangular array or table of numbers. We call the individual numbers entriesof the matrix and refer to them by their row and column numbers. The rows are numbered1;2;:::from the top and the columns are numbered1;2;:::from left to right.
So we use what you might think of as a(row, colum) coordinate system for the entries of a matrix.In the example2
41 1 2 5
1 11 132
2 1 3 43
513 is the(2;3)entry, the entry in row 2 and column 3.
The matrix above is called a34matrix because it has 3 rows and 4 columns. We can talk about matrices of all different sizes such as 4 5 7 11 224 7 21
4 7 122
44 5
7 11
13 133
5 32and in general we can havemnmatrices for anym1andn1. Matrices with just one row are calledrow matrices. A1nmatrix[x1x2xn]has just the same information in it as ann-tuple(x1;x2;:::;xn)2Rnand so we could be tempted to identify1nmatrices withn-tuples (which we know are points or vectors inRn). We use the termcolumn matrixfor a matrix with just one column. Here is ann1(column) matrix 2 6 664x
1 x 2... x n3 7 775
and again it is tempting to think of these as the "same" asn-tuples(x1;x2;:::;xn)2Rn. Maybe not quite as tempting as it is for row matrices, but not such a very different idea. Toavoidconfusionthatwouldcertainlyariseifweweretomakeeitheroftheseidentifications (either of1nmatrices withn-tuplesorofn1matrices withn-tuples) we will not make either
of them and keep all the different objects in their own separate places. A bit later on, it will often
be more convenient to think of columnn1matrices as points ofRn, but we will not come to that for some time.22012-13 Mathematics MA1S11 (Timoney)
Now, to clarify any confusion these remarks might cause, we explain that we consider two matrices to be the 'same" matrix only if they are absolutely identical. They have to have the same shape (same number of rows and same number of columns) and they have to have the same numbers in the same positions. Thus, all the following are different matrices 1 2 3 4 6=2 1 3 46=2 1 0
3 4 0 2 42 13 4 0 03 5
3.2 Double subscripts
When we want to discuss a matrix without listing the numbers in it, that is when we want to discuss a matrix that is not yet specified or an unknown matrix we use a notation like this with double subscriptsx11x12 x 21x22This is a22matrix where the(1;1)entry isx11, the(1;2)entry isx12and so on. It would probably be clearer of we put commas in and write x1;1x1;2 x
2;1x2;2
instead, but people commonly use the version without the commas between the two subscripts. Carrying this idea further, when we want to discuss anmnmatrixXand refer to its entries we write X=2 6 664x11x12x1n
x21x22x2n.........
x m1xm2xmn3 7 775So the(i;j)entry ofXis calledxij. (It might be more logical to call the matrixxin lower case, and the entriesxijas we have done, but it seems more common to use capital letters lineXfor matrices.) Sometimes we want to write something like this but we don"t want to take up space for the whole picture and we write an abbreviated version like
X= [xij]1im;1jn
To repeat what we said about when matrices are equal using this kind of notation, suppose we have twomnmatricesX= [xij]1im;1jnandY= [yij]1im;1jn
ThenX=Ymeans themnscalar equationsxij=yijmust all hold (for each(i;j)with1im;1jn). And if anmnmatrix equals anrsmatrix, we have to havem=r
(same number or rows),n=s(same number of columns) and then all the entries equal.Matrices3
3.3 Arithmetic with matrices
In much the same way as we did withn-tuples we now define addition of matrices. We only allow addition of matrices that are of the same size. Two matrices of different sizes cannot be added.If we take twomnmatrices
X= [xij]1im;1jnandY= [yij]1im;1jn
then we defineX+Y= [xij+yij]1im;1jn
(themnmatrix with(1;1)entry the sum of the(1;1)entries ofXandY,(1;2)entry the sum of the(1;2)entries ofXandY, and so on).For example
2 42 134
0 73 5 +2 462
15 12 9 213 5 =2
42 + 6 1 + (2)
3 + 154 + 12
0 + (9) 7 + 213
5 =2 48118 8 9 283 5 We next define the scalar multiplekX, for a numberkand a matrixX. We just multiply every entry ofXbyk. So if
X= [xij]1im;1jn
is anymnmatrix andkis any real number thenkXis anothermnmatrix. Specifically kX= [kxij]1im;1jnFor example For example
8 2 42 134
0 73 5 =2
48(2) 8(1)
8(3) 8(4)
8(0) 8(7)3
5 =2 416 82432
0 563 5 We see that if we multiply byk= 0we get a matrix where all the entries are 0. This has a special name. Themnmatrix where every entry is 0 is called themnzero matrix. Thus we have zero matrices of every possible size.