6) Hello please help me with this urgently

Question

Let A be an n x n matrix and consider the linear homogeneous system Ax = 0. If the linear system has only the trivial solution state whether the following statements are true or false.

(a) 0 is an eigenvalue of A.

(b) All columns of A are basic columns.

(c) Rank of A is n.

Answer 1

Hi Arwen: if a matrix equation has only the trivial solution, ie, that x={0,0,0...}, it must be possible to put the matrix into reduced row echelon form so that there's a diagonal of 1s and everything else 0, ie, the identity matrix. So for a 3x3, we'd see (I hope this displays ok):

|1 0 0| | x | | 0 |

|0 1 0| | y | = | 0 |

|0 0 1| | z | | 0 |

and the trivial solution would be x=0, y=0, z=0.

Now when a matrix is in diagonal form, the eigenvalues are all neatly along the main diagonal and we see they are only 1's. No zeroes. So (a) is false.

A basic column is a column with a pivot (ask me more if you need to know what a pivot is). Every single column here has a pivot. So (b) is true.

Finally, this matrix that I drew has a rank of 3 and an nxn matrix would have a rank of n. So (c) is true.

Ask if you need more explanations of any of these things.