Question

Suppose ADequalsISubscript m ​(the mtimesm identity​ matrix). Show that for any Bold b in set of real numbers R Superscript m​, the equation ABold xequalsBold b has a solution.​ [Hint: Think about the equation ADBold bequalsBold b​.] Explain why A cannot have more rows than columns.

Answer 1

Answer:

See Explanation

Step-by-step explanation:

(a)For matrices A and D, given that: $AD=I_m$.

We want to show that $\forall b \in R^m$, Ax=b has a solution.

If Ax=b

Multiply both sides by D

$(Ax)D=b \times D\\\implies (AD)x=bD (Recall: AD=I_m)\\\implies I_mx=Db (Since I_m is the m\times m identity matrix)\\\implies x=Db$

This means that the system Ax=b has a solution.

(b)Matrix A has a pivot position in each row where each pivot is a different column. Therefore, A must have at least as many columns as rows.

This means A cannot have more rows than columns.