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,  ${\text{det }}A$  is a scalar multiple of  ${\text{det }}B.$
So,
${\text{det }}A=0$
and the statement is false.

Final Answer:
FALSE

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