 Hello and welcome to the session in this session. We will discuss about mattresses First of all, let's define a matrix a Matrix whose plural is mattresses is Any array of real numbers or Some other suitable entities arranged in rows and columns So basically we say that any rectangular array of numbers is called a Matrix this is a rectangular array of numbers. This is a matrix We generally use the large parenthesis to identify a matrix and we can even use brackets for matrix and also Double vertical bars are also used in matrix the horizontal lines of numbers That is these horizontal lines these are called rows Then the vertical lines of numbers that is these are the vertical lines of numbers and These are called columns Also, each number or entity in a matrix is called an element or entry of the matrix So if you consider this matrix the numbers 2, 3, 4, 5, 6, and 7 Are the elements or you can say the entries of this matrix We denote matrix by the capital letters like a b c and so on and The entries or you can say the elements of Matrix and denoted by small letters a b c and so on So if you consider this matrix, let this be denoted by letter a Let's now discuss order of a matrix. A matrix has n columns when we say of the matrix is n by n or You can say a matrix with m rows in columns is n by n Matrix while stating the order of a matrix We first state the number of rows and then the number of columns If you consider this matrix a with elements 2, 4, 6, 5, 7, 9 Then we find that in this matrix a there are three columns the matrix a is 2 by 3 That is we first take the number of rows and then the number of columns So we say that this matrix a is a 2 by 3 Matrix next we have special types of matrices First is a matrix having only one row is the No matrix Consider a matrix B The elements 2, 3, 4 This is a 1 by 3 matrix That is it has one row and three columns One by three is the order of this matrix B As you can see that there is just one row in this matrix. So we say That this B that is the matrix B is a Row matrix next is column matrix having only One column is a column matrix consider a matrix C with elements 2, 3, 4 This is a 3 by 1 matrix that is there are three rows and one column Now as it is only one column in this matrix. So we say that this matrix C Is a column matrix Next square matrix a matrix having Same number of rows two columns a square matrix consider a matrix D With elements 2, 3, 4, 5 Now this is a 2 by 2 matrix that is there are two rows and two columns in this matrix Which means that the number of rows and the number of columns in this matrix are equal. So this is a Square matrix now as the number of rows and number of columns For this square matrix D is 2 so we can also denote this as B2 which means That in this square matrix the number of rows and number of columns are 2 next we have a Zero matrix a matrix all elements zero a zero matrix or a Null matrix a zero or null matrix is generally denoted by O now consider a 2 by 3 zero matrix That is in which we have two rows three columns in this matrix all the elements are zero So this is a null matrix or a zero matrix Next we have Dienal matrix a square matrix and elements zero Except the principal diagonal elements a diagonal matrix For example consider a matrix X or three by three order in which the elements are three zero zero zero one zero Zero zero four this is a three by three matrix Now the principal diagonal means the diagonal from the top left to the top bottom so This should you to see Principal diagonal whose elements are three one four As in this matrix X Which is a square matrix all the elements are zero except these principal diagonal elements These principal diagonal elements Some of these diagonal elements can be zero, but all of them cannot be zero Next we have a unit matrix or Identity matrix a square matrix each diagonal element as unity and Other elements as zero is called a unit matrix or identity Matrix a unit matrix of order n is denoted by I n like if you have I 2 This is a unit matrix of order two that is we have two rows and two columns in this and It has elements one run in the diagonal and rest of the elements are zero Let's now discuss the general form of a matrix say a matrix a Now the general form of this matrix is given by a i j of order m by n or We can simply write this as a matrix with elements a i j We have Order of the matrix a is M by n that is n is the number of rows n is the number of columns element a i j of the matrix it belongs to the ith row the jth column and this is called i j element the matrix a And in this element a i j of the matrix a The subscript i it represents the row number and the subscript j represents the column number so The element say a 1 2 means that this element belongs to The first row second Column that is one represents the row number and two represents the column number We can also write the general form of the matrix a as the matrix with elements in the first row as a 1 1 a 1 2 a 1 3 and so on up to a 1 n since we have n number of columns In the matrix a Then the elements in the second row are given by a 2 1 a 2 2 a 2 3 and so on up to a 2 n In the same day, we will proceed for the rest of the rows as there are m rows in the matrix a so the last row of The matrix a could be given as a m 1 a m 2 a m 3 and so on up to a Mn so this is a matrix a of order m by n That is there are m rows and n columns now suppose consider a matrix B With elements in the first row as six two in the second row as three minus one and in the third row as zero five So this matrix B is of order three by two that is there are three rows and two columns now in this The element a 1 1 that is in the first row and the first column then element 2 is 1 2 that is the element appearing in the first row and the second column then a 2 1 is the element appearing in the second row and first column which is three a 2 2 is the element appearing in the second row and second column and this is minus one then a 3 1 is the element appearing in the third row and first column and this is zero then Then a 3 2 is the element appearing in the third row and second column which is five Next we discuss the equality of matrices two matrices are equal and only if They satisfy the two conditions that is both matrices are of the same order and Where corresponding elements are equal With elements in the first row as two one zero and in the second row the elements are three two One tricks B with elements in the first row as six by three three by three Zero by three in the second row the elements are three four minus two and two minus one Now both the matrices a and b are of order two by three So they satisfy the first condition that is both the matrices should be of same order Now let's see if the corresponding elements are equal Now two is equivalent to six by three one is equivalent to three by three Zero is equivalent to zero by three Three is equivalent to three this two is equivalent to four minus two and one is equivalent to two minus one So the corresponding elements are also equal. This means that the matrices a and b are equal Now let's consider a matrix C with elements In the first row as two one in the second row the elements three three four Or consider a matrix D with elements In the first row as one two in the second row as four three Now the matrices C and D both are of order two by two But the corresponding elements are not equal So they do not satisfy both the conditions of equality of matrices So this means the matrix C is not equal to the matrix D This completes the session. Hope you understood the concept of matrices and the special types of matrices