031 Review Part 1 - Revision history

Kayla Murray at 02:34, 10 October 2017

2017-10-10T02:34:17Z

Kayla Murray at 20:06, 9 October 2017

2017-10-09T20:06:18Z

Kayla Murray: Created page with "'''This is a sample, and is meant to represent the material usually covered in Math 9C for the final. An actual test may or may not be similar.''' '''Click on the
2017-10-09T18:09:28Z

Created page with "'''This is a sample, and is meant to represent the material usually covered in Math 9C for the final. An actual test may or may not be similar.''' '''Click on the <span class..."

New page
'''This is a sample, and is meant to represent the material usually covered in Math 9C for the final. An actual test may or may not be similar.'''

'''Click on the  boxed problem numbers  to go to a solution.'''
<div class="noautonum">TOC</div>

== [[009C_Sample Final 1,_Problem_1| Problem 1 ]] ==
True or false: If all the entries of a  <math style="vertical-align: 0px">7\times 7</math>  matrix  <math style="vertical-align: 0px">A</math>  are  <math style="vertical-align: -4px">7,</math>  then  <math style="vertical-align: 0px">\text{det }A</math>  must be  <math style="vertical-align: 0px">7^7.</math>

== [[009C_Sample Final 1,_Problem_2| Problem 2 ]] ==
 True or false: If a matrix  <math style="vertical-align: 0px">A^2</math>  is diagonalizable, then the matrix  <math style="vertical-align: 0px">A</math>  must be diagonalizable as well.

== [[009C_Sample Final 1,_Problem_3| Problem 3 ]] ==
True or false: If  <math style="vertical-align: 0px">A</math>  is a  <math style="vertical-align: -1px">4\times 4</math>  matrix with characteristic equation  <math style="vertical-align: -5px">\lambda(\lambda-1)(\lambda+1)(\lambda+e)=0,</math>  then  <math style="vertical-align: 0px">A</math>  is diagonalizable.

== [[009C_Sample Final 1,_Problem_4| Problem 4 ]] ==
 True or false: If  <math style="vertical-align: 0px">A</math>  is invertible, then  <math style="vertical-align: 0px">A</math>  is diagonalizable.

== [[009C_Sample Final 1,_Problem_5| Problem 5 ]] ==
True or false: If  <math style="vertical-align: 0px">A</math>  and  <math style="vertical-align: 0px">B</math>  are invertible  <math style="vertical-align: 0px">n\times n</math>  matrices, then so is  <math style="vertical-align: -1px">A+B.</math>

== [[009C_Sample Final 1,_Problem_6| Problem 6 ]] ==
 True or false: If  <math style="vertical-align: 0px">A</math>  is a  <math style="vertical-align: 0px">3\times 5</math>  matrix and  <math style="vertical-align: -4px">\text{dim Nul }A=2,</math>  then  <math style="vertical-align: 0px">A\vec{x}=\vec{b}</math>  is consistent for all  <math style="vertical-align: 0px">\vec{b}</math>  in  <math style="vertical-align: 0px">\mathbb{R}^3.</math>

== [[009C_Sample Final 1,_Problem_7| Problem 7 ]] ==

True or false: Let  <math style="vertical-align: 0px">C=AB</math>  for  <math style="vertical-align: 0px">4\times 4</math>  matrices  <math style="vertical-align: 0px">A</math>  and  <math style="vertical-align: 0px">B.</math>  If  <math style="vertical-align: 0px">C</math>  is invertible, then  <math style="vertical-align: 0px">A</math>  is invertible.

== [[009C_Sample Final 1,_Problem_8| Problem 8 ]] ==

True or false: Let  <math style="vertical-align: 0px">W</math>  be a subspace of  <math style="vertical-align: 0px">\mathbb{R}^4</math>  and  <math style="vertical-align: 0px">\vec{v}</math>  be a vector in  <math style="vertical-align: 0px">\mathbb{R}^4.</math>  If  <math style="vertical-align: 0px">\vec{v}\in W</math>  and  <math style="vertical-align: -4px">\vec{v}\in W^\perp,</math>  then  <math style="vertical-align: 0px">\vec{v}=\vec{0}.</math>

== [[009C_Sample Final 1,_Problem_9| Problem 9 ]] ==

True or false: If  <math style="vertical-align: 0px">A</math>  is an invertible  <math style="vertical-align: 0px">3\times 3</math>  matrix, and  <math style="vertical-align: 0px">B</math>  and  <math style="vertical-align: 0px">C</math>  are  <math style="vertical-align: 0px">3\times 3</math>  matrices such that  <math style="vertical-align: -4px">AB=AC,</math>  then  <math style="vertical-align: 0px">B=C.</math>

← Older revision		Revision as of 02:34, 10 October 2017
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−	'''~~This is a~~ sample, and is meant to represent the material usually covered in Math 9C for the final. An actual test may or may not be similar.'''	+	'''These questions are from sample exams and actual exams at other universities. The questions are meant to represent the material usually covered in Math 31 for the final. An actual test may or may not be similar.'''

	'''Click on the <span class="biglink" style="color:darkblue;"> boxed problem numbers </span> to go to a solution.'''		'''Click on the <span class="biglink" style="color:darkblue;"> boxed problem numbers </span> to go to a solution.'''

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 <div class="noautonum">__TOC__</div>
-== [[009C_Sample Final 1,_Problem_1|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 1&nbsp;</span></span>]] ==
+== [[031_Review Part 1,_Problem_1|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 1&nbsp;</span></span>]] ==
 <span class="exam">True or false: If all the entries of a &nbsp;<math style="vertical-align: 0px">7\times 7</math>&nbsp; matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; are &nbsp;<math style="vertical-align: -4px">7,</math>&nbsp; then &nbsp;<math style="vertical-align: 0px">\text{det }A</math>&nbsp; must be &nbsp;<math style="vertical-align: 0px">7^7.</math>
-== [[009C_Sample Final 1,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
+== [[031_Review Part 1,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
 <span class="exam"> True or false: If a matrix &nbsp;<math style="vertical-align: 0px">A^2</math>&nbsp; is diagonalizable, then the matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; must be diagonalizable as well.
-== [[009C_Sample Final 1,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
+== [[031_Review Part 1,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
 <span class="exam">True or false: If &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is a &nbsp;<math style="vertical-align: -1px">4\times 4</math>&nbsp; matrix with characteristic equation &nbsp;<math style="vertical-align: -5px">\lambda(\lambda-1)(\lambda+1)(\lambda+e)=0,</math>&nbsp; then &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is diagonalizable.
-== [[009C_Sample Final 1,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
+== [[031_Review Part 1,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
 <span class="exam"> True or false: If &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is invertible, then &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is diagonalizable.
-== [[009C_Sample Final 1,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
+== [[031_Review Part 1,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
 <span class="exam">True or false: If &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">B</math>&nbsp; are invertible &nbsp;<math style="vertical-align: 0px">n\times n</math>&nbsp; matrices, then so is &nbsp;<math style="vertical-align: -1px">A+B.</math>
-== [[009C_Sample Final 1,_Problem_6|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 6&nbsp;</span>]] ==
+== [[031_Review Part 1,_Problem_6|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 6&nbsp;</span>]] ==
 <span class="exam"> True or false: If &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is a &nbsp;<math style="vertical-align: 0px">3\times 5</math>&nbsp; matrix and &nbsp;<math style="vertical-align: -4px">\text{dim Nul }A=2,</math>&nbsp; then &nbsp;<math style="vertical-align: 0px">A\vec{x}=\vec{b}</math>&nbsp; is consistent for all &nbsp;<math style="vertical-align: 0px">\vec{b}</math>&nbsp; in &nbsp;<math style="vertical-align: 0px">\mathbb{R}^3.</math>
-== [[009C_Sample Final 1,_Problem_7|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 7&nbsp;</span>]] ==
+== [[031_Review Part 1,_Problem_7|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 7&nbsp;</span>]] ==
 <span class="exam">True or false: Let &nbsp;<math style="vertical-align: 0px">C=AB</math>&nbsp; for &nbsp;<math style="vertical-align: 0px">4\times 4</math>&nbsp; matrices &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">B.</math>&nbsp; If &nbsp;<math style="vertical-align: 0px">C</math>&nbsp; is invertible, then &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is invertible.
-== [[009C_Sample Final 1,_Problem_8|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 8&nbsp;</span>]] ==
+== [[031_Review Part 1,_Problem_8|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 8&nbsp;</span>]] ==
 <span class="exam">True or false: Let &nbsp;<math style="vertical-align: 0px">W</math>&nbsp; be a subspace of &nbsp;<math style="vertical-align: 0px">\mathbb{R}^4</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">\vec{v}</math>&nbsp; be a vector in &nbsp;<math style="vertical-align: 0px">\mathbb{R}^4.</math>&nbsp; If &nbsp;<math style="vertical-align: 0px">\vec{v}\in W</math>&nbsp; and &nbsp;<math style="vertical-align: -4px">\vec{v}\in W^\perp,</math>&nbsp; then &nbsp;<math style="vertical-align: 0px">\vec{v}=\vec{0}.</math>
-== [[009C_Sample Final 1,_Problem_9|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 9&nbsp;</span>]] ==
+== [[031_Review Part 1,_Problem_9|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 9&nbsp;</span>]] ==
 <span class="exam">True or false: If &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is an invertible &nbsp;<math style="vertical-align: 0px">3\times 3</math>&nbsp; matrix, and &nbsp;<math style="vertical-align: 0px">B</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">C</math>&nbsp; are &nbsp;<math style="vertical-align: 0px">3\times 3</math>&nbsp; matrices such that &nbsp;<math style="vertical-align: -4px">AB=AC,</math>&nbsp; then &nbsp;<math style="vertical-align: 0px">B=C.</math>