Question

7. Which of the following statements isn't true? Explain your reasoning. (Hint: There is only one false statement.) (a) If the columns of an n x n matrix form a basis of R", then the matrix will be invertible. (b) If A is invertible, then A-1 is also invertible. (c) If A is an n xn matrix whose columns span R", then A must be one-to-one. (d) If A is an n x n matrix, then the preimage of the zero vector is always just the zero vector. (e) If the columns of an n x n matrix are linearly independent, then the rows are also linearly independent.

Answer 1

7 @ True In an nx n matrix there are n columns if they form a basis of 1Rm, it means columus are linearly independent, and since совет отм а таа зраи , A so rank (A) = n : A is onto. => A is one-to-one and onto (as by Rank nullity Theorem) fence - A is invertible

☺ Tree eet A is invertible then gat such I= ATA that I -> . AAT= A is the At is also AT inverse of invertible. Hence True:- f n columus spaus irm n must be linearly independent are linearly dependent they cannot It means rank (A) = n By Rank Wallity Theoren then all as if they generate IR". (A) rank + nullity (A) = n . . . @ m + n(A) = m n(A0 . 5) Ker (A) = zero space Hence A is one-to-one. False :- of A is an na ni matrix then preimage q zero is not always the zero vector for example take 2x2 matrix ( 27 [24] Here columus are linearly dependent so not onto means not one-tor me. so the preimage of zero vector are many like {}] [ +3 = 1:] ind [ 17 [?]=[:] (~2,1) and (-4,2) are the preimage of zero.

© The True - the columns are linearly independent Then 1 column (rank = n. Row (rank) = coluum (rank) an . .- . n linearly independent rows. rence Rows are also linearly independent. conclusion the is that only (d) is the false statement : The refore