1. (16 points) In the following A in always an ?NVERTIBLE \" following is true o
ID: 3282972 • Letter: 1
Question
1. (16 points) In the following A in always an ?NVERTIBLE " following is true or falen (a) A is row equivaleut to the ideutity n tnatrix. Ine-shel True Falsn b) A mst be diagonalizable True False (e) A hos a pivot in every column False (d) det A 0 True (e) 0 cannot be an eigeuvalue of A True False (0) 1 must be an eigenvalue of A True False (g) The columns of A span R True False (h) The equation Ax - bis always consistent for every b in R" True False 2, (8 points) In the following A is an m × n matrix. (a) If Ax O has only the trivial solution then the columns of A are linearly indep True False n. (b) If Ax = b is always consistent for every b in R", then m False True (c) The rank of A is equal to n. False True (d) The rank of AT equals the rank of A. False TrueExplanation / Answer
Understand all steps and any doubt in this comment below. I will explain...
1- A be n*n invertible matrix... So it's eigen value and determinant never be zero .... (d) and (e) is true....
Also rows and columns span R^n... And row and column is equivalent to identity... So (a) , (c) and (g) be true...
Also AX =B is consistent if A is invertible.. so (h) is true..
Since matrix A is not given.. so it is not sure that 1 is eigen value of A or A is diagonalizable... So (b) and (f) is false...
2 - A be m*n matrix.. so it have m equations and n variables...
So if rank of A is equal to n then it has consistent i.e. columns are linearly independent... So (a) is true.
AX = B is not always consistent because its depends on matrix A but no informations given about rank of matrix A.. so (b) is false.. .. also (c) is false...
Generally rank(A) = rank(A^t) .. so (d) is true..
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.