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\"><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mn>11</mn></mtd><mtd><mn>6</mn></mtd></mtr><mtr><mtd><mn>9</mn></mtd><mtd><mn>5</mn></mtd></mtr></mtable></mfenced><mo> </mo><mi>and</mi><mo> </mo><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mn>5</mn></mtd><mtd><mo>-</mo><mn>6</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>9</mn></mtd><mtd><mn>11</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/638badad48ca6dd6-1682602478281.jpg "open square brackets table row 11 6 row 9 5 end table close square brackets space and space open square brackets table row 5 cell negative 6 end cell row cell negative 9 end cell 11 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>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>2</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo> </mo><mi>and</mi><mo> </mo><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/f24342ea1ccf1ebc-1682602680930.jpg "open square brackets table row 1 0 cell negative 3 end cell row cell negative 1 end cell 2 0 row 1 0 cell negative 1 end cell end table close square brackets space and space open square brackets table row 0 1 0 row 0 1 1 row cell negative 3 end cell cell negative 2 end cell 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><mo>-</mo><mn>3</mn></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>5</mn></mtd><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/ac39595a175c9a91-1682604831540.jpg "open square brackets table row cell negative 3 end cell 5 row cell negative 5 end cell 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><mo>-</mo><mn>5</mn></mtd><mtd><mo>-</mo><mn>7</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/39041d0e0c8eba3f-1682605012404.jpg "open square brackets table row cell negative 5 end cell cell negative 7 end cell row 2 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>8</mn></mtd><mtd><mn>6</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>12</mn></mtd><mtd><mo>-</mo><mn>9</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/a8c325225c8f3d07-1682605030114.jpg "open square brackets table row 8 6 row cell negative 12 end cell cell negative 9 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>3</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>3</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>7</mn></mtd><mtd><mn>3</mn></mtd><mtd><mn>3</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/c794028272b29677-1682609580765.jpg "open square brackets table row 3 0 3 row 0 cell negative 3 end cell 0 row 7 3 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>4</mn></mtd><mtd><mn>5</mn></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>7</mn></mtd><mtd><mn>6</mn></mtd><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>7</mn></mtd><mtd><mn>5</mn></mtd><mtd><mn>6</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/54e34f89a3f19387-1682609992551.jpg "open square brackets table row 4 5 5 row 7 6 5 row 7 5 6 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>8</mn></mtd><mtd><mo>-</mo><mn>12</mn></mtd></mtr><mtr><mtd><mn>8</mn></mtd><mtd><mn>4</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>10</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/bd215ee3c200fa54-1682611628352.jpg "open square brackets table row 8 8 cell negative 12 end cell row 8 4 0 row cell negative 1 end cell cell negative 1 end cell 10 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><msqrt><mn>2</mn></msqrt></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>3</mn><msqrt><mn>3</mn></msqrt></mtd><mtd><mn>4</mn><msqrt><mn>7</mn></msqrt></mtd></mtr></mtable></mfenced><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><msqrt><mn>2</mn></msqrt></mtd><mtd><mo>-</mo><msqrt><mn>10</mn></msqrt></mtd></mtr><mtr><mtd><mn>5</mn><msqrt><mn>3</mn></msqrt></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/535e91f6bc35c818-1682423777351.jpg "open square brackets table row cell square root of 2 end cell 1 row cell 3 square root of 3 end cell cell 4 square root of 7 end cell end table close square brackets open square brackets table row cell square root of 2 end cell cell negative square root of 10 end cell row cell 5 square root of 3 end cell 0 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><msqrt><mn>5</mn></msqrt></mtd><mtd><msqrt><mn>5</mn></msqrt></mtd><mtd><mo>-</mo><msqrt><mn>20</mn></msqrt></mtd></mtr><mtr><mtd><msqrt><mn>7</mn></msqrt></mtd><mtd><msqrt><mn>343</mn></msqrt></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mn>12</mn></mtd><mtd><mo>-</mo><mn>14</mn></mtd></mtr><mtr><mtd><mn>7</mn></mtd><mtd><mn>16</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>4</mn></mtd></mtr></mtable></mfenced><mspace linebreak=\"newline\"></mspace></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/b693ddd1c5085963-1682423755130.jpg "open square brackets table row cell square root of 5 end cell cell square root of 5 end cell cell negative square root of 20 end cell row cell square root of 7 end cell cell square root of 343 end cell 0 end table close square brackets open square brackets table row 12 cell negative 14 end cell row 7 16 row 0 4 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>2</mn></mtd><mtd><mo>-</mo><mn>7</mn></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>6</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>13</mn></mtd></mtr></mtable></mfenced><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>9</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/3094c036b10b3a34-1682423133560.jpg "open square brackets table row 2 cell negative 7 end cell 3 row 6 0 13 end table close square brackets open square brackets table row cell negative 2 end cell row 5 row 9 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>5</mn></mtd><mtd><mn>7</mn></mtd></mtr><mtr><mtd><mn>6</mn></mtd><mtd><mn>11</mn></mtd></mtr></mtable></mfenced><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/df3f7b6801042180-1682422950691.jpg "open square brackets table row 5 7 row 6 11 end table close square brackets open square brackets table row cell negative 2 end cell row 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\"><mi>A</mi><mo>=</mo><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mo>-</mo><mn>5</mn></mtd><mtd><mn>9</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>11</mn></mtd></mtr></mtable></mfenced><mo> </mo><mi>and</mi><mo> </mo><mi>B</mi><mo>=</mo><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mo>-</mo><mn>12</mn></mtd><mtd><mn>6</mn></mtd></mtr><mtr><mtd><mn>9</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/886d2b1e53f88d0a-1682420984007.jpg "A equals open square brackets table row cell negative 5 end cell 9 row 0 11 end table close square brackets space and space B equals open square brackets table row cell negative 12 end cell 6 row 9 0 end table close square brackets")
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![{"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>5</mn></mtd><mtd><mo>-</mo><mn>8</mn></mtd><mtd><mo>-</mo><mn>10</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>6</mn></mtd><mtd><mo>-</mo><mn>7</mn></mtd><mtd><mn>9</mn></mtd></mtr></mtable></mfenced><mfenced open=\"[\" close=\"]\"><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>2</mn></mtd><mtd><mn>9</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>5</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd><mtd><mn>7</mn></mtd><mtd><mo>-</mo><mn>8</mn></mtd></mtr></mtable></mfenced></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/72c317e910494d82-1682424701602.jpg "open square brackets table row cell negative 5 end cell cell negative 8 end cell cell negative 10 end cell row 3 cell negative 1 end cell 0 row 6 cell negative 7 end cell 9 end table close square brackets open square brackets table row 0 2 9 row 2 5 cell negative 2 end cell row 4 7 cell negative 8 end cell end table close square brackets")