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Exercise 4.1

Evaluate the determinants in Exercise 1 and 2.

<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>2</mn></mtd><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>5</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Solun</mi><mo mathvariant="bold">:</mo><mo mathvariant="bold">-</mo><mo mathvariant="bold">&#xA0;</mo><mi>Let</mi><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>2</mn></mtd><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>5</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>2</mn><mo>&#xD7;</mo><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>4</mn><mo>&#xD7;</mo><mfenced><mrow><mo>-</mo><mn>5</mn></mrow></mfenced><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mo>-</mo><mn>2</mn><mo>+</mo><mn>20</mn><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>18</mn><mspace linebreak="newline"/></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mo>&#xA0;</mo><mfenced><mi mathvariant="normal">i</mi></mfenced><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi>cos</mi><mo>&#xA0;</mo><mi mathvariant="normal">&#x3B8;</mi></mtd><mtd><mo>-</mo><mi>sin</mi><mo>&#xA0;</mo><mi mathvariant="normal">&#x3B8;</mi></mtd></mtr><mtr><mtd><mi>sin</mi><mo>&#xA0;</mo><mi mathvariant="normal">&#x3B8;</mi></mtd><mtd><mi>cos</mi><mo>&#xA0;</mo><mi mathvariant="normal">&#x3B8;</mi></mtd></mtr></mtable></mfenced></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Solun</mi><mo mathvariant="bold">:</mo><mo mathvariant="bold">-</mo><mo>&#xA0;</mo><mi>Let</mi><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi>cos</mi><mo>&#xA0;</mo><mi mathvariant="normal">&#x3B8;</mi></mtd><mtd><mo>-</mo><mi>sin</mi><mo>&#xA0;</mo><mi mathvariant="normal">&#x3B8;</mi></mtd></mtr><mtr><mtd><mi>sin</mi><mo>&#xA0;</mo><mi mathvariant="normal">&#x3B8;</mi></mtd><mtd><mi>cos</mi><mo>&#xA0;</mo><mi mathvariant="normal">&#x3B8;</mi></mtd></mtr></mtable></mfenced><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mi>cos</mi><mo>&#xA0;</mo><mi mathvariant="normal">&#x3B8;</mi><mo>&#xD7;</mo><mi>cos</mi><mo>&#xA0;</mo><mi mathvariant="normal">&#x3B8;</mi><mo>-</mo><mi>sin</mi><mo>&#xA0;</mo><mi mathvariant="normal">&#x3B8;</mi><mo>&#xD7;</mo><mfenced><mrow><mo>-</mo><mi>sin</mi><mo>&#xA0;</mo><mi mathvariant="normal">&#x3B8;</mi></mrow></mfenced><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><msup><mi>cos</mi><mn>2</mn></msup><mo>&#xA0;</mo><mi mathvariant="normal">&#x3B8;</mi><mo>+</mo><msup><mi>sin</mi><mn>2</mn></msup><mo>&#xA0;</mo><mi mathvariant="normal">&#x3B8;</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>1</mn></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>ii</mi></mfenced><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mtd><mtd><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mtd><mtd><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Solun</mi><mo mathvariant="bold">:</mo><mo mathvariant="bold">-</mo><mo mathvariant="bold">&#xA0;</mo><mi>Let</mi><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mtd><mtd><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mtd><mtd><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced><mrow><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><msup><mi mathvariant="normal">x</mi><mn>3</mn></msup><mo>+</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mi mathvariant="normal">x</mi><mo>+</mo><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>+</mo><mi mathvariant="normal">x</mi><mo>-</mo><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><msup><mi mathvariant="normal">x</mi><mn>3</mn></msup><mo>-</mo><mo>&#xA0;</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mo>&#xA0;</mo><mi>If</mi><mo>&#xA0;</mo><mi mathvariant="normal">A</mi><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mo>&#xA0;</mo><mi>then</mi><mo>&#xA0;</mo><mi>show</mi><mo>&#xA0;</mo><mi>that</mi><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>2</mn><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>4</mn><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Solun</mi><mo mathvariant="bold">:</mo><mo mathvariant="bold">-</mo><mo mathvariant="bold">&#xA0;</mo><mi>Given</mi><mo>&#xA0;</mo><mi mathvariant="normal">A</mi><mo>=</mo><msub><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mrow><mn>2</mn><mo>&#xD7;</mo><mn>2</mn></mrow></msub><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mn>2</mn><mi mathvariant="normal">A</mi><mo>=</mo><mn>2</mn><msub><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mrow><mn>2</mn><mo>&#xD7;</mo><mn>2</mn><mo>&#xA0;</mo></mrow></msub><mo>=</mo><mo>&#xA0;</mo><msub><mfenced open="[" close="]"><mtable><mtr><mtd><mn>2</mn></mtd><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>8</mn></mtd><mtd><mn>4</mn></mtd></mtr></mtable></mfenced><mrow><mn>2</mn><mo>&#xD7;</mo><mn>2</mn></mrow></msub><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>2</mn><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>2</mn></mtd><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>8</mn></mtd><mtd><mn>4</mn></mtd></mtr></mtable></mfenced><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>2</mn><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>2</mn></mtd><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>8</mn></mtd><mtd><mn>4</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>8</mn><mo>-</mo><mn>32</mn><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mo>-</mo><mn>24</mn></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Taking</mi><mo>&#xA0;</mo><mi mathvariant="normal">R</mi><mo>.</mo><mi mathvariant="normal">H</mi><mo>.</mo><mi mathvariant="normal">S</mi><mo>.</mo><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mn>4</mn><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>4</mn><mfenced open="|" close="|"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>4</mn><mfenced><mrow><mn>2</mn><mo>-</mo><mn>8</mn></mrow></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mo>-</mo><mn>24</mn></math>

So, |2A| = 4|A|   (Hence Proved………)

<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mo>&#xA0;</mo><mi>If</mi><mo>&#xA0;</mo><mi mathvariant="normal">A</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>4</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>,</mo><mo>&#xA0;</mo><mi>then</mi><mo>&#xA0;</mo><mi>show</mi><mo>&#xA0;</mo><mi>that</mi><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>3</mn><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>27</mn><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced></math>    

<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Solun</mi><mo mathvariant="bold">:</mo><mo mathvariant="bold">-</mo><mo mathvariant="bold">&#xA0;</mo><mi>Given</mi><mo>&#xA0;</mo><mi mathvariant="normal">A</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><msub><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>4</mn></mtd></mtr></mtable></mfenced><mrow><mn>3</mn><mo>&#xD7;</mo><mn>3</mn></mrow></msub><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mn>3</mn><mi mathvariant="normal">A</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><msub><mfenced open="[" close="]"><mtable><mtr><mtd><mn>3</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>3</mn></mtd><mtd><mn>6</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>12</mn></mtd></mtr></mtable></mfenced><mrow><mn>3</mn><mo>&#xD7;</mo><mn>3</mn></mrow></msub><mspace linebreak="newline"/><mo>&#x21D2;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>3</mn><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>3</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>3</mn></mtd><mtd><mn>6</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>12</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mfenced><mrow><mi>Expand</mi><mo>&#xA0;</mo><mi>by</mi><mo>&#xA0;</mo><msub><mi mathvariant="normal">R</mi><mn>1</mn></msub></mrow></mfenced><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>3</mn><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>3</mn><mfenced><mrow><mn>36</mn><mo>-</mo><mn>0</mn></mrow></mfenced><mo>-</mo><mn>0</mn><mfenced><mrow><mn>0</mn><mo>-</mo><mn>0</mn></mrow></mfenced><mo>+</mo><mn>3</mn><mfenced><mrow><mn>0</mn><mo>-</mo><mn>0</mn></mrow></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>108</mn></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x21D2;</mo><mo>&#xA0;</mo><mn>27</mn><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>27</mn><mfenced open="|" close="|"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>4</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mfenced><mrow><mi>Expand</mi><mo>&#xA0;</mo><mi>by</mi><mo>&#xA0;</mo><msub><mi mathvariant="normal">R</mi><mn>1</mn></msub></mrow></mfenced><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mn>27</mn><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>27</mn><mfenced open="[" close="]"><mrow><mn>1</mn><mfenced><mrow><mn>4</mn><mo>-</mo><mn>0</mn></mrow></mfenced><mo>-</mo><mn>0</mn><mfenced><mrow><mn>0</mn><mo>-</mo><mn>0</mn></mrow></mfenced><mo>+</mo><mn>1</mn><mfenced><mrow><mn>0</mn><mo>-</mo><mn>0</mn></mrow></mfenced></mrow></mfenced><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mn>27</mn><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>27</mn><mo>&#xD7;</mo><mn>4</mn><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>108</mn></math>

5. Evaluate the determinants

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="normal">i</mi></mfenced><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>5</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Solun</mi><mo mathvariant="bold">:</mo><mo mathvariant="bold">-</mo><mo mathvariant="bold">&#xA0;</mo><mi>Let</mi><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>5</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mfenced><mrow><mi>Expand</mi><mo>&#xA0;</mo><mi>by</mi><mo>&#xA0;</mo><msub><mi mathvariant="normal">R</mi><mn>1</mn></msub></mrow></mfenced><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>3</mn><mfenced><mrow><mn>0</mn><mo>-</mo><mn>5</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>0</mn><mo>+</mo><mn>3</mn></mrow></mfenced><mo>-</mo><mn>2</mn><mfenced><mrow><mn>0</mn><mo>-</mo><mn>0</mn></mrow></mfenced><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mo>-</mo><mn>15</mn><mo>+</mo><mn>3</mn><mo>+</mo><mn>0</mn><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mo>-</mo><mn>12</mn></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>ii</mi></mfenced><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>4</mn></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>3</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Solun</mi><mo mathvariant="bold">:</mo><mo mathvariant="bold">-</mo><mo>&#xA0;</mo><mi>Let</mi><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>4</mn></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>3</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>3</mn><mfenced><mrow><mn>1</mn><mo>+</mo><mn>6</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mo>-</mo><mn>4</mn></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><mn>4</mn></mrow></mfenced><mo>+</mo><mn>5</mn><mfenced><mrow><mn>3</mn><mo>-</mo><mn>2</mn></mrow></mfenced><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>21</mn><mo>+</mo><mn>20</mn><mo>+</mo><mn>5</mn><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>46</mn></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>iii</mi></mfenced><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd><mtd><mn>3</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Solun</mi><mo mathvariant="bold">:</mo><mo mathvariant="bold">-</mo><mo>&#xA0;</mo><mi>Let</mi><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd><mtd><mn>3</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>0</mn><mfenced><mrow><mn>0</mn><mo>+</mo><mn>9</mn></mrow></mfenced><mo>-</mo><mn>1</mn><mfenced><mrow><mn>0</mn><mo>-</mo><mn>6</mn></mrow></mfenced><mo>+</mo><mn>2</mn><mfenced><mrow><mo>-</mo><mn>3</mn><mo>-</mo><mn>0</mn></mrow></mfenced><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>0</mn><mo>+</mo><mn>6</mn><mo>-</mo><mn>6</mn><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>0</mn></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>iv</mi></mfenced><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>2</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>2</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>5</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Solun</mi><mo mathvariant="bold">:</mo><mo mathvariant="bold">-</mo><mo>&#xA0;</mo><mi>Let</mi><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>2</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>2</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>5</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>2</mn><mfenced><mrow><mn>0</mn><mo>-</mo><mn>5</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>0</mn><mo>+</mo><mn>3</mn></mrow></mfenced><mo>+</mo><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>0</mn><mo>-</mo><mn>6</mn></mrow></mfenced><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mo>-</mo><mn>10</mn><mo>+</mo><mn>3</mn><mo>+</mo><mn>12</mn><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mn>5</mn></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mo>&#xA0;</mo><mi>If</mi><mo>&#xA0;</mo><mi mathvariant="normal">A</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd><mtd><mn>4</mn></mtd><mtd><mo>-</mo><mn>9</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>,</mo><mo>&#xA0;</mo><mi>find</mi><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Solun</mi><mo mathvariant="bold">:</mo><mo mathvariant="bold">-</mo><mo mathvariant="bold">&#xA0;</mo><mi>Given</mi><mo mathvariant="bold">&#xA0;</mo><mi mathvariant="normal">A</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><msub><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd><mtd><mn>4</mn></mtd><mtd><mo>-</mo><mn>9</mn></mtd></mtr></mtable></mfenced><mrow><mn>3</mn><mo>&#xD7;</mo><mn>3</mn></mrow></msub><mspace linebreak="newline"/><mo>&#x21D2;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">A</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd><mtd><mn>4</mn></mtd><mtd><mo>-</mo><mn>9</mn></mtd></mtr></mtable></mfenced></math>

|A| = 1(-9+12)-1(-18+15)-2(8-5)

|A| = 0

7. Find values of x, if

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="normal">i</mi></mfenced><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>2</mn></mtd><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>2</mn><mi>x</mi></mtd><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>6</mn></mtd><mtd><mi>x</mi></mtd></mtr></mtable></mfenced></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Solun</mi><mo mathvariant="bold">:</mo><mo mathvariant="bold">-</mo><mo mathvariant="bold">&#xA0;</mo><mi>Given</mi><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>2</mn></mtd><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>2</mn><mi mathvariant="normal">x</mi></mtd><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>6</mn></mtd><mtd><mi mathvariant="normal">x</mi></mtd></mtr></mtable></mfenced></math>

2-20 = 2x2-24

⇒ -18 = 2x2-24

⇒ 2x2 = 24-18 = 6

⇒ x2 = 3

⇒ x = ±3

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>ii</mi></mfenced><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>2</mn></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd><mtd><mn>5</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi>x</mi></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn><mi>x</mi></mtd><mtd><mn>5</mn></mtd></mtr></mtable></mfenced><mspace linebreak="newline"/><mi mathvariant="bold">Solun</mi><mo mathvariant="bold">:</mo><mo mathvariant="bold">-</mo><mo mathvariant="bold">&#xA0;</mo><mi>Given</mi><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>2</mn></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd><mtd><mn>5</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">x</mi></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn><mi mathvariant="normal">x</mi></mtd><mtd><mn>5</mn></mtd></mtr></mtable></mfenced></math>

10-12 = 5x-6x

⇒ -2 = -x

⇒ x = 2

<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>.</mo><mo>&#xA0;</mo><mi>If</mi><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">x</mi></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>18</mn></mtd><mtd><mi mathvariant="normal">x</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>6</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>18</mn></mtd><mtd><mn>6</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>,</mo><mo>&#xA0;</mo><mi>then</mi><mo>&#xA0;</mo><mi mathvariant="normal">x</mi><mo>&#xA0;</mo><mi>is</mi><mo>&#xA0;</mo><mi>equal</mi><mo>&#xA0;</mo><mi>to</mi></math>

(A) 6     (B) ±6     (C) -6     (D) 0

<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold">Solun</mi><mo mathvariant="bold">:</mo><mo mathvariant="bold">-</mo><mo mathvariant="bold">&#xA0;</mo><mi>Given</mi><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mi mathvariant="normal">x</mi></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>18</mn></mtd><mtd><mi mathvariant="normal">x</mi></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>6</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>18</mn></mtd><mtd><mn>6</mn></mtd></mtr></mtable></mfenced></math>

⇒ x2 - 36 = 36 - 36

⇒ x2 = 36

⇒ x = ±6

Answer is ……… B.


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