031 Review Problems - Revision history

Kayla Murray at 20:11, 25 August 2017

2017-08-25T20:11:46Z

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Kayla Murray at 19:24, 25 August 2017

2017-08-25T19:24:51Z

Kayla Murray at 19:12, 25 August 2017

2017-08-25T19:12:22Z

Kayla Murray at 18:33, 25 August 2017

2017-08-25T18:33:04Z

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Kayla Murray at 01:13, 25 August 2017

2017-08-25T01:13:16Z

Kayla Murray at 01:00, 25 August 2017

2017-08-25T01:00:25Z

Kayla Murray at 00:53, 25 August 2017

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Kayla Murray at 00:36, 25 August 2017

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Kayla Murray at 00:23, 25 August 2017

2017-08-25T00:23:52Z

Kayla Murray at 00:20, 25 August 2017

2017-08-25T00:20:14Z

@@ Line 47: / Line 47: @@
 |}
-'''4.''' True or false: If &nbsp;<math>A</math>&nbsp; is invertible, then &nbsp;<math>A</math>&nbsp; is diagonalizable.
+'''4.''' 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.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 62: / Line 62: @@
 |}
-'''5.''' True or false: If &nbsp;<math>A</math>&nbsp; and &nbsp;<math>B</math>&nbsp; are invertible &nbsp;<math>n\times n</math>&nbsp; matrices, then so is &nbsp;<math>A+B.</math>
+'''5.''' 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>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 77: / Line 77: @@
 |}
-'''6.''' True or false: If &nbsp;<math>A</math>&nbsp; is a &nbsp;<math>3\times 5</math>&nbsp; matrix and &nbsp;<math>\text{dim Nul }A=2,</math>&nbsp; then &nbsp;<math>A\vec{x}=\vec{b}</math>&nbsp; is consistent for all &nbsp;<math>\vec{b}</math>&nbsp; in &nbsp;<math>\mathbb{R}^3.</math>
+'''6.''' 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>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 92: / Line 92: @@
 |}
-'''7.''' True or false: Let &nbsp;<math>C=AB</math>&nbsp; for &nbsp;<math>4\times 4</math>&nbsp; matrices &nbsp;<math>A</math>&nbsp; and &nbsp;<math>B.</math>&nbsp; If &nbsp;<math>C</math>&nbsp; is invertible, then &nbsp;<math>A</math>&nbsp; is invertible.
+'''7.''' 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.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 107: / Line 107: @@
 |}
-'''8.''' True or false: Let &nbsp;<math>W</math>&nbsp; be a subspace of &nbsp;<math>\mathbb{R}^4</math>&nbsp; and &nbsp;<math>\vec{v}</math>&nbsp; be a vector in &nbsp;<math>\mathbb{R}^4.</math>&nbsp; If &nbsp;<math>\vec{v}\in W</math>&nbsp; and &nbsp;<math>\vec{v}\in W^\perp,</math>&nbsp; then &nbsp;<math>\vec{v}=\vec{0}.</math>
+'''8.''' 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>
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@@ Line 122: / Line 122: @@
 |}
-'''9.''' True or false: If &nbsp;<math>A</math>&nbsp; is an invertible &nbsp;<math>3\times 3</math>&nbsp; matrix, and &nbsp;<math>B</math>&nbsp; and &nbsp;<math>C</math>&nbsp; are &nbsp;<math>3\times 3</math>&nbsp; matrices such that &nbsp;<math>AB=AC,</math>&nbsp; then &nbsp;<math>B=C.</math>
+'''9.''' 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>
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@@ Line 139: / Line 139: @@
 '''10.'''
-(a) Is the matrix &nbsp;<math>A=
+(a) Is the matrix &nbsp;<math style="vertical-align: -18px">A=
      \begin{bmatrix}
 & 1 \\
@@ Line 145: / Line 145: @@
           \end{bmatrix}</math>&nbsp; diagonalizable? If so, explain why and diagonalize it. If not, explain why not.
-(b) Is the matrix &nbsp;<math>A=
+(b) Is the matrix &nbsp;<math style="vertical-align: -31px">A=
      \begin{bmatrix}
 & 0 & -2 \\
@@ Line 165: / Line 165: @@
 |}
-'''11.''' Find the eigenvalues and eigenvectors of the matrix &nbsp;<math>A=
+'''11.''' Find the eigenvalues and eigenvectors of the matrix &nbsp;<math style="vertical-align: -31px">A=
      \begin{bmatrix}
 & 1 & 1 \\

@@ Line 2: / Line 2: @@
-'''1.''' True or false: If all the entries of a &nbsp;<math>7\times 7</math>&nbsp; matrix &nbsp;<math>A</math>&nbsp; are &nbsp;<math>7,</math>&nbsp; then &nbsp;<math>\text{det }A</math>&nbsp; must be &nbsp;<math>7^7.</math>
+'''1.''' 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>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 17: / Line 17: @@
 |}
-'''2.''' True or false: If a matrix &nbsp;<math>A^2</math>&nbsp; is diagonalizable, then the matrix &nbsp;<math>A</math>&nbsp; must be diagonalizable as well.
+'''2.''' 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.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 32: / Line 32: @@
 |}
-'''3.''' True or false: If &nbsp;<math>A</math>&nbsp; is a &nbsp;<math>4\times 4</math>&nbsp; matrix with characteristic equation &nbsp;<math>\lambda(\lambda-1)(\lambda+1)(\lambda+e)=0,</math>&nbsp; then &nbsp;<math>A</math>&nbsp; is diagonalizable.
+'''3.''' 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.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"

@@ Line 459: / Line 459: @@
 |}
-'''21.'''
+'''21.''' Let <math>A=
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 473: / Line 479: @@
 |}
-'''22.'''
+'''22.''' Find a formula for <math>\begin{bmatrix}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 488: / Line 498: @@
 '''23.'''
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 501: / Line 516: @@
 |}
-'''24.'''
+'''24.''' Let <math>A=\begin{bmatrix}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 515: / Line 545: @@
 |}
-'''25.'''
+'''25.''' Give an example of a <math>3\times 3</math> matrix <math>A</math> with eigenvalues 5,-1 and 3.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 529: / Line 560: @@
 |}
-'''26.'''
+'''26.''' Assume <math>A^2=I.</math> Find Nul <math>A.</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 543: / Line 575: @@
 |}
-'''27.'''
+'''27.''' If <math>A</math> is an <math>n\times n</math> matrix such that <math>AA^T=I,</math> what are the possible values of det <math>A?</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 557: / Line 590: @@
 |}
-'''28.'''
+'''28.''' Show that if <math>\vec{x}</math> is an eigenvector of the matrix product <math>AB</math> and <math>B\vec{x}\ne \vec{0},</math> then <math>B\vec{x}</math> is an eigenvector of <math>BA.</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 572: / Line 606: @@
 '''29.'''
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 585: / Line 624: @@
 |}
-'''30.'''
+'''30.''' Consider the following system of equations.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 599: / Line 645: @@
 |}
-'''31.'''
+'''31.''' Suppose <math>\{\vec{u},\vec{v}\}</math> is a basis of the eigenspace corresponding to the eigenvalue 0 of a <math>5\times 5</math> matrix <math>A.</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;

@@ Line 336: / Line 336: @@
 |}
-'''17.'''
+'''17.''' Let <math>A=
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 350: / Line 359: @@
 |}
-'''18.'''
+'''18.''' Let <math>\vec{v}=\begin{bmatrix}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 364: / Line 388: @@
 |}
-'''19.'''
+'''19.''' Let <math>W=</math>Span<math>\Bigg\{\begin{bmatrix}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 379: / Line 419: @@
 '''20.'''
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;

@@ Line 1: / Line 1: @@
 '''This is a list of sample problems and is meant to represent the material usually covered in Math 31. An actual test may or may not be similar.'''
 '''1.''' True or false: If all the entries of a <math>7\times 7</math> matrix <math>A</math> are <math>7,</math> then det <math>A</math> must be <math>7^7.</math>
@@ Line 16: / Line 17: @@
 |}
-'''2.'''
+{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Solution:
+!Final Answer: &nbsp;
 |-
+|
 |}
+'''3.''' True or false: If <math>A</math> is a <math>4\times 4</math> matrix with characteristic equation <math>\lambda(\lambda-1)(\lambda+1)(\lambda+e)=0,</math> then <math>A</math> is diagonalizable.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Solution:
+!Solution: &nbsp;
 |-
+|
 |}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Solution:
+!Final Answer: &nbsp;
 |-
+|
 |}
-'''7.'''
+'''4.''' True or false: If <math>A</math> is invertible, then <math>A</math> is diagonalizable.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Solution:
+!Solution: &nbsp;
 |-
+|
 |}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Solution:
+!Final Answer: &nbsp;
 |-
+|
 |}