031 Review Part 1, Problem 6

True or false: If  $A$  is a  $3\times 5$  matrix and  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\text{dim Nul }}A=2,}   then  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A{\vec {x}}={\vec {b}}}   is consistent for all  ${\vec {b}}$  in  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {R} ^{3}.}

Solution:
By the Rank Theorem, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {5}&=&\displaystyle {{\text{dim Col }}A+{\text{dim Nul }}A}\\&&\\&=&\displaystyle {{\text{dim Col }}A+2.}\end{array}}}
Hence,  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\text{dim Col }}A=3.}
This tells us that  $A$  has three pivots.
Since  $A$  is a  $3\times 5$  matrix,
$A$  has a pivot in every row.
Therefore,  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A{\vec {x}}={\vec {b}}}   is consistent for all  ${\vec {b}}$  in  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {R} ^{3}.}
So, the statement is true.

Final Answer:
TRUE

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