Difference between revisions of "031 Review Part 1, Problem 1"

Revision as of 15:00, 9 October 2017

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

@@ Line 4: / Line 4: @@
 !Solution: &nbsp;
 |-
-|First, we switch to the limit to <math style="vertical-align: 0px">x</math> so that we can use L'Hopital's rule.
+|If all the entries of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; are &nbsp;<math style="vertical-align: -4px">7,</math>&nbsp; then all the rows of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; are identical.
 |-
-|So, we have
+|So, when you row reduce &nbsp;<math style="vertical-align: -4px">A,</math>&nbsp; it is row equivalent to a matrix &nbsp;<math style="vertical-align: -4px">B,</math>&nbsp; where &nbsp;<math style="vertical-align: 0px">B</math>&nbsp; contains a row of zeros.
+|-
+|Then,
 |-
 |
-&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
+::<math>\text{det } B=0.</math>
-\displaystyle{\lim_{x \rightarrow \infty}\frac{3-2x^2}{5x^2 + x +1}} & \overset{L'H}{=} & \displaystyle{\lim_{x \rightarrow \infty}\frac{-4x}{10x+1}}\\
+|-
-&&\\
+|But, &nbsp;<math style="vertical-align: -1px">\text{det }A</math>&nbsp; is a scalar multiple of &nbsp;<math style="vertical-align: -1px">\text{det }B.</math>&nbsp;
-& \overset{L'H}{=} & \displaystyle{\frac{-4}{10}}\\
+|-
-&&\\
+|So,
-& = & \displaystyle{-\frac{2}{5}}.
+|-
-\end{array}</math>
+|
+::<math>\text{det }A=0</math>
+|-
+|and the statement is false.
+|-
 |}
@@ Line 21: / Line 27: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp;&nbsp; &nbsp; &nbsp; False
+|&nbsp;&nbsp; &nbsp; &nbsp; FALSE
 |}
 [[031_Review_Part_1|'''<u>Return to Sample Exam</u>''']]