031 Review Part 1, Problem 1

True or false: If all the entries of a  $7\times 7$  matrix  $A$  are  $7,$  then  ${\text{det }}A$  must be  $7^{7}.$

Solution:
If all the entries of  $A$  are  $7,$  then all the rows of  $A$  are identical.
So, when you row reduce  $A,$  it is row equivalent to a matrix  $B,$  where  $B$  contains a row of zeros.
Then,
${\text{det }}B=0.$
But,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{det }A}   is a scalar multiple of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{det }B.}
So,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{det }A=0}
and the statement is false.

Final Answer:
FALSE

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