Algebra and Trigonometry Unit 3 Session 1

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[00:00:09]my name is Christoph Yao goddessa I hold enfields in mathematics education I'll Take You Through Math 101.

[00:00:24]algebra and trigonometry we are on unit 3 which is operations on matrices and we will start with section one which is metric let's look at the objectives of section one by the end of this section you should be able to one describe a mattress by indicating the number of columns and rows it has to give examples of special metrics let's consider some definitions a mattress usually has M Rule and N column enclosed in parentheses the numbers in the parentheses are called the elements of the Matrix horizontally There are rules and vertically there are column so to mention the Matrix we start with the rule and then we follow it up with the columns so here is example one we talk about a matrix that is two by three two by three here means that the rows are two and the columns are three so you look at the Matrix and you see that Row one is made up of two three and one row two is made up of five seven and four column one is two five column two is three seven and column three is one four The Matrix has six entry uh elements in general an M by n Matrix has emerald and N column here is a matrix of size three by three the entries two one and four lie on the main icon of the Matrix the other diagonal is the skew diagonal so there are two diagonals when it comes every Matrix the main diagonal is two one four then the other diagonal which is made up of seven one five is what we call the scale diagonal there could be many rows and many columns in a given Matrix so a general representation of a matrix say a is what you are seeing there which is a one one a one a one three up to a one n A1 and because all the ones we have mentioned so far are in row one so you will see that they start with one each of them starts with one one one one two one three one four one five and so on then the second one a two one eight two two a two three up to a two n so that tells you that we are talking about row two and so the first entries are always always two then you can also talk about the columns the First Column is a one one then the second one is a two one and the third one is a three one you see that the second entry is always one so that tells you that you are talking about column one when you go to the second one you see that the second entry is always two so a one two a two two a two three so that tells you that we are talking about column two and so on so the first when you take a particular entry the first index talks about the rule the second one talks about the the column so if you take a two three that tells you that that element is in Row 2 column three if you take a five four means that the element is in row five column four so an example a12 that means that you have the elements in row one column two of Matrix a let's talk about some special matrices the first one we'll talk about is the unit Matrix it is also called the identity Matrix it is a square Matrix whose only non-zero elements are on the diagonal and are equal to one so you see that the main diagonal for that matrices one one all other entries are zero zero that is for three by three you can also have a four by four unit Matrix where the first row will be one zero zero zero the second row will be zero one zero zero the third one will be zero zero one zero and the fourth one zero zero zero one another special Matrix to be considered is the zero Med and for the zero Matrix each element in the parenthesis is equal to zero you can have it for three by three you can have two by two you can even have four by four a two by two first row will be zero zero the second row will be zero zero five three by three the first row will be zero zero the second will be zero zero zero and the third will be zero zero zero for a four by four the first row will be zero zero zero zero second row zero zero zero zero third row zero zero zero zero and the fourth row zero zero zero zero so that is the zero Matrix we can also talk about the column Matrix and the column Matrix such that you have only one column but there are three you can also have one that there are two rules or one that there are four rows but one column so if you look at the example here Matrix B is equal to seven three so this is where you have one column but you have three rules you can also have another one where the rows are say two but the column is one you can also have another one where the rows are four but the column is still one just as we have the column Matrix we also have a row metric and in the row Matrix there is only one row it is only the column that varies so for example Matrix C is equal to three one nine in this example we have one rule but three columns you can also have a situation where the row is one The Columns are trying to have another one where the row is still one but the columns are four another special Matrix we can look at is the diagonal matrix and the diagonal matrix is such that the main diagonal has none zero element was the rest of the entries are zero for example if you look at the Matrix D the main diagonal is made up of 21 9 and negative three but the rest of the numbers are zero the next thing to look at is what we call Matrix equality for Matrix equality two matrices are equal if they have the same size and if they are corresponding elements are identical so a is equal to B if and only a a i j is equal to b i j for I is equal to 1 up to M and J is equal to 1 up to n so what this means is that for any two matrices to be equal the order of the two matrices Mass first of all be the same and then the number of elements must be the same and then the type of element the kind of elements must also be the same so if you have a set a which is one two three four that is two by two one first row three four second row and then B must also be first row one two three four two by two for you to conclude that Matrix a is equal to Matrix B let's find the transpose of a matrix the transpose of a matrix is obtained by interchanging Edge rows and eight columns this means that if you have for example a three by three Matrix and you want to find eight transport the first row of the Matrix becomes the First Column the second row of that Matrix becomes the second column and the third row becomes the third column let's consider the Matrix a which is a one one a one a one three up to a one n and the rest of them at the next stage we'll find a transpose of this Matrix the transpose of the Matrix given previously is presented here so you see a superscript t that indicates the transcript of a and you see that Row one in that Matrix has become column one in this Matrix row two of that Matrix has become column two of these Matrix and so on let's look at the example find the transpose of the Matrix m is equal to nine seven three one eight five seven four can you quickly work this and get your answer down before we get to the solution so you can see whether you are getting it or not here is the solution to the previous question I guess you've gotten it right you'll see that Row one of the Matrix which was 9 7 has now become column one of these new Matrix 297 and row two which was three one eight has now become column two three one eight and Row three which was five seven four is now column one sorry is now column three which is five seven four now the Matrix transpose satisfies some rules rule one if you find the transpose of a matrix and then you find the transpose of the answer you come back to the original Matrix two if you add two matrices that are of the same Dimension I find the transpose of the answer it is going to be the same as finding the transpose of each of them separately and then adding them the same thing happens with multiplication you multiply the two matrices you find the transpose it's the same as doing the transpose of each of them before multiplying and remember the two matrices must have compatible dimensions [Music]