Question

Suppose CAequalsISubscript n ​(the ntimesn identity​ matrix). Show that the equation ABold xequalsBold 0 has only the trivial solution. Explain why A cannot have more columns than rows

Answer 1

See Explanation

Explanation:

(a)For matrices A and C, given that:
`CA=I_n`.

We want to show that Ax=0 has only the trivial solution

If Ax=0

Multiply both sides by C

`C(Ax)=C * 0\\\implies (CA)x=0$ (Recall: CA=I_n)\\\implies I_nx=0 $ (Since I_n$ is the n* n$ identity matrix)\\\implies x=0`

This means that the system has only the trivial solution.

(b)If the system has more columns than rows, a free variable would occur when a column does not have a pivot. This would lead to a non-trivial solution.