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Nana76 [90]
3 years ago
5

Suppose CAequalsISubscript n ​(the ntimesn identity​ matrix). Show that the equation ABold xequalsBold 0 has only the trivial so

lution. Explain why A cannot have more columns than rows
Mathematics
1 answer:
ruslelena [56]3 years ago
7 0

Answer:

See Explanation

Step-by-step 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 \times 0\\\implies (CA)x=0$ (Recall: CA=I_n)\\\implies I_nx=0 $ (Since I_n$ is the n\times 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.

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