#CSIRNET CSIR NET MATHEMATICS NOV 2020 | 30th Nov | Section C | Linear Algebra | Rank of a Matrix

Hello friends, this is Mahesh Shivalekar and you are watching NextGen. If you have not subscribed my channel, then please subscribe the channel and press the bell icon. In this video, we are going to discuss the problem of CSIRON 8, November 2020. Problem of Linear Algebra. Question Let A be an m x n matrix. Let A1 colon, A colon 1, and A11 be the matrices obtained from A by deleting row 1, deleting column 1, deleting both row 1 and column 1 respectively, which of following holds. We have given four options. First option, rank of A minus 2 less than or equal to rank of A11 less than or equal to rank of A. Second option, rank of A1 colon equal to rank of A colon 1. Third option, rank of A equal to rank of A1 column equal to rank of A colon 1 than rank of A equal to rank of A11. Fourth option, rank of A1 colon plus rank of A colon 1 plus 2 greater than or equal to 2 times rank of A. If you want notes of this solution then please visit our website www.nextgenmaths.com In the description link is available It is a problem of rank of a matrix So first we recall how can we find the rank of a matrix The number of independent rows or columns will be the rank of a matrix That means in the matrix the number of independent row or number of independent column will give the rank of a matrix C Here given three definitions First definition A1 column is the matrix obtained from matrix A by deleting row 1 A column 1 is the matrix obtained from A by deleting column 1 And A11 is the matrix obtained from A by deleting both row 1 and column 1 If we delete a row from the matrix A then rank of a matrix reduced by maximum number 1 That will give the rank of A-1 less than equal to rank of A1 column less than equal to rank of A Rank of A1 column is obtained by deleting row 1 That means rank of a matrix reduced by maximum 1 With the same argument if we delete any column from matrix A Then rank of a matrix A reduced by maximum 1 So we have rank of A-1 less than equal to rank of A column 1 less than equal to rank of A A11 is obtained by deleting 1 row and 1 column That means rank of A reduced by maximum number 2 So we have rank of A-2 less than equal to rank of A11 less than equal to rank of A So from this three definitions we have three inequalities If we add first two inequalities then we have rank of A plus rank of A equal to 2 rank of A minus 1 minus 1 equal to minus 2 less than equal to rank of A1 column plus rank of A column 1 less than equal to 2 times rank of A If we consider the first part of inequality then we have 2 times rank of A less than equal to rank of A1 column plus rank of A column 1 plus 2 From inequality number 3 and inequality number 4 We can say that the option 1 and option 4 are the correct answer Now we are going to discuss the option number 2 In which we have given that rank of A1 column equal to rank of A column 1 Just consider matrix A with this entry 0 1 0 0 0 0 0 0 0 A1 column is a matrix obtained by deleting first row So if we delete first row from the matrix A then we have 0 0 0 0 0 0 And matrix A column 1 is obtained by deleting first column of the matrix That will give the 1 0 0 0 0 0 So rank of A1 column is 0 and rank of A column 1 is 1 That means they are not equal But option number 2 says that it is always equal which is wrong So option number 2 is a wrong answer Option number 3 In the option number 3 we have given that rank of A equal to rank of A1 column equal to rank of A column 1 That means there is no reduction in the rank of A by deleting row 1 or column 1 In other words first row and first column are not independent row and independent column of a matrix A If they are independent then rank of matrix is reduced by 1 But we have given that rank of A equal to rank of A1 column equal to rank of A column 1 So they do not contribute anything in the rank of matrix A That means if we delete row 1 and column 1 from the matrix A There is no reduction in the rank of matrix A That means rank of matrix A remains same after deleting row 1 and column 1 Therefore rank of A equal to rank of A11 which is obtained by deleting first row and first column So option number 3 is the correct answer Thanks for watching For more videos like, share, subscribe, press the bell icon of NextGenMS