7. Systems of Equations & Matrices
Determinants and Cramer's Rule
7. Systems of Equations & Matrices
Determinants and Cramer's Rule: Videos & Practice Problems
Determinants and Cramer's Rule Practice Problems
136 problems
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![{"mathml":"<math style=\"font-family:stix;font-size:16px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"16px\"><mi>A</mi><mo>=</mo><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mo>-</mo><mn>6</mn></mtd><mtd><mo>-</mo><mn>5</mn></mtd></mtr><mtr><mtd><mn>7</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mo> </mo><mi>B</mi><mo>=</mo><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mn>8</mn></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>4</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>9</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mo> </mo><mi>C</mi><mo>=</mo><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd><mtd><mn>7</mn></mtd><mtd><mn>8</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>4</mn></mtd><mtd><mn>12</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/1c91be8628399e07-1682440604210.jpg "A equals open square brackets table row cell negative 6 end cell cell negative 5 end cell row 7 2 end table close square brackets comma space B equals open square brackets table row 8 5 row 0 cell negative 4 end cell row 1 9 end table close square brackets comma space C equals open square brackets table row cell negative 2 end cell 7 8 row 0 4 12 end table close square brackets")
![{"mathml":"<math style=\"font-family:stix;font-size:16px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"16px\"><mi>A</mi><mo>=</mo><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mn>7</mn></mtd><mtd><mn>8</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>5</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mo> </mo><mi>B</mi><mo>=</mo><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mn>10</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>4</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/bdc47b75f71a9522-1682442381268.jpg "A equals open square brackets table row 7 8 row cell negative 5 end cell 1 end table close square brackets comma space B equals open square brackets table row 10 0 row 3 cell negative 4 end cell end table close square brackets")
![{"mathml":"<math style=\"font-family:stix;font-size:16px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"16px\"><mi>A</mi><mo>=</mo><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>9</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd><mtd><mn>6</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mo> </mo><mi>B</mi><mo>=</mo><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mn>7</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>8</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/bd87e4b26103f3c3-1682443451426.jpg "A equals open square brackets table row 0 cell negative 9 end cell row cell negative 3 end cell 6 end table close square brackets comma space B equals open square brackets table row 7 cell negative 2 end cell row cell negative 1 end cell 8 end table close square brackets")
![{"mathml":"<math style=\"font-family:stix;font-size:16px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"16px\"><mi>A</mi><mo>=</mo><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mi>p</mi></mtd><mtd><mi>q</mi></mtd></mtr><mtr><mtd><mi>r</mi></mtd><mtd><mi>s</mi></mtd></mtr></mtable></mfenced><mo> </mo><mi>and</mi><mo> </mo><mi>B</mi><mo>=</mo><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/45387fb07d118a50-1682502583153.jpg "A equals open square brackets table row p q row r s end table close square brackets space and space B equals open square brackets table row 1 0 row 0 1 end table close square brackets")
![{"mathml":"<math style=\"font-family:stix;font-size:16px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"16px\"><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mn>4</mn></mtd><mtd><mi>p</mi><mo>+</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>11</mn><mi>r</mi></mtd><mtd><mi>s</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mi>q</mi></mtd><mtd><mn>3</mn><mi>p</mi><mo>-</mo><mn>5</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>4</mn><mi>r</mi></mtd><mtd><mn>7</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/bb3a88fa09252f25-1682502748409.jpg "open square brackets table row 4 cell p plus 3 end cell row cell negative 11 r end cell s end table close square brackets equals open square brackets table row q cell 3 p minus 5 end cell row cell negative 4 r end cell 7 end table close square brackets")
![{"mathml":"<math style=\"font-family:stix;font-size:16px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"16px\"><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>6</mn></mtd></mtr></mtable></mfenced><mo>-</mo><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mn>9</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>5</mn></mtd></mtr><mtr><mtd><mn>7</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mn>8</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/f85d2b8a80ec9fbb-1682438424546.jpg "open square brackets table row 4 row 1 row 6 end table close square brackets minus open square brackets table row 9 row cell negative 5 end cell row 7 end table close square brackets plus open square brackets table row 8 row 0 row 3 end table close square brackets")
![{"mathml":"<math style=\"font-family:stix;font-size:16px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"16px\"><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mn>7</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mn>6</mn></mtd><mtd><mn>11</mn></mtd></mtr></mtable></mfenced><mo>-</mo><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mn>9</mn></mtd><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>8</mn></mtd><mtd><mn>3</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/d1a748b0806c334e-1682438528104.jpg "open square brackets table row 7 1 5 row 3 6 11 end table close square brackets minus open square brackets table row 9 4 row 8 3 end table close square brackets")
![{"mathml":"<math style=\"font-family:stix;font-size:16px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"16px\"><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd><mtd><mn>7</mn></mtd></mtr></mtable></mfenced><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mo>-</mo><mn>8</mn></mtd><mtd><mn>7</mn></mtd></mtr><mtr><mtd><mn>9</mn></mtd><mtd><mn>13</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/627ed9defdd3f4d3-1682438580051.jpg "open square brackets table row cell negative 2 end cell 0 row 4 7 end table close square brackets open square brackets table row cell negative 8 end cell 7 row 9 13 end table close square brackets")
![{"mathml":"<math style=\"font-family:stix;font-size:16px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"16px\"><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mn>8</mn></mtd><mtd><mn>1</mn></mtd><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>6</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>7</mn></mtd></mtr></mtable></mfenced><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mo>-</mo><mn>4</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>7</mn></mtd><mtd><mn>5</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/c4bb1397086d74cc-1682438786570.jpg "open square brackets table row 8 1 cell negative 3 end cell row 6 0 7 end table close square brackets open square brackets table row cell negative 4 end cell 0 row 0 4 row 7 5 end table close square brackets")
![{"mathml":"<math style=\"font-family:stix;font-size:16px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"16px\"><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mo>-</mo><mn>3</mn></mtd><mtd><mn>7</mn></mtd><mtd><mn>7</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>2</mn></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>9</mn></mtd><mtd><mo>-</mo><mn>6</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mn>3</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/72ff3ea9f929fc16-1682438975993.jpg "open square brackets table row cell negative 3 end cell 7 7 row 0 2 5 row 9 cell negative 6 end cell cell negative 2 end cell end table close square brackets open square brackets table row 3 0 cell negative 3 end cell row cell negative 3 end cell 0 0 row 3 3 cell negative 3 end cell end table close square brackets")
![{"mathml":"<math style=\"font-family:stix;font-size:16px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"16px\"><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mn>8</mn></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>11</mn></mtd><mtd><mn>7</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/ea1c698bb11b4235-1682502998132.jpg "open square brackets table row 8 5 row 11 7 end table close square brackets")
![{"mathml":"<math style=\"font-family:stix;font-size:16px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"16px\"><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>4</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd><mtd><mo>-</mo><mn>3</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/71e531984d2008e5-1682503097070.jpg "open square brackets table row 3 cell negative 4 end cell 0 row 4 0 4 row 4 cell negative 3 end cell 0 end table close square brackets")