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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.

User Mibollma
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Answer:

See Explanation

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 * D\\\implies (AD)x=bD$ (Recall: AD=I_m)\\\implies I_mx=Db $ (Since I_m$ is the m* 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.

User Fariya Rahmat
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