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Of course, we will start with the linear problems initially and after that we will go to the non-linear and other type of problems also. Now, let us give some introduction as I told initially to the basic mathematical tools which will be required to us. You may know everything, but in brief we will tell the things. What is required or which we will be used afterwards in this lecture. (Refer Slide Time: 10:41) The first one is matrix algebra. In matrix algebra actually if you see, if you have m cross n matrix. This m cross n matrix is we say, is an arrangement of (m n) objects. This

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objects may be distinct may not be distinct in and you are placing this (m n) objects in m rows and n columns in a particular form. Then we call it as a matrix. So, whenever we are defining this m cross n matrix is basically nothing but; a arrangement of (m n) objects in m rows and n columns and we write it in the form like this. a 1 1 a 1 2 a 1 n, a 2 1 a 2 2 a 2 n, a m 1 a m 2 a m n. So, we have total (m n) objects. We say that the matrix is of order m cross n since, it has m rows and n columns. So, it is order is m cross n. And whenever m is not equals n we tell that the matrix as rectangular matrix. Now, each object a i j basically, these objects a i j is we call it as an element of the matrix. That is the element on the i th row and on the j th column. This element we call it as the element of the matrix. Whereas, if you see a 1 1 a 2 2 like this way a n n that is this diagonals a 1 1 a 2 2 a m m sorry, a m n. This one we call it as the diagonal elements. Now, Row matrix, just some definitions; a matrix of order1 cross n, if I write down a matrix of order 1 cross n, that is 1 row and n columns. Then we call it as the row matrix. And we write it as a 1 1 a 1 2 like this way a 1 n. So, this one is a row matrix. Similarly, you can define the column matrix; in case of column matrix what happens, where we have a matrix which takes the form of m cross 1 that is m rows and 1 column. Then we call it as the column matrix and it takes the form of this one; a 1 1 a 2 1 like this way it will go up to a m 1. So, a 1 1 a 2 1 a m 1 this one we are calling as the column matrix. (Refer Slide Time: 14:37)
Next comes, square matrix; if you have a matrix of order m cross n, where m equals n. That is number of rows and numbers of columns are equal. Then we call that matrix as a square matrix of order n. When ever, you are having the number of rows and numbers of columns are same then we call that matrix as the square matrix. We can write down like this. So, this matrix is the square matrix which has n rows and n columns. Next here comes, null matrix; you have the matrix A equals a i j, the order is m cross n. If a i j equals 0 for all a i and j then that matrix we call it as the null matrix. I can write down something like this 0 0 0, 0 0 0 and all are 0.So, this matrix whatever I written is a null matrix of order 3 or it has 3 rows and 3 columns.

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• Spring '16
• Sameh

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