Question
3.4: Applications of the Determinant
Videos
Cramer’s Rule
Find the Area of a Triangle on the Coordinate Plane Using a Determinant
Determine if a
Let
Determinant of matrix
Matrix
Submit Ourstion
Videos
Cramer’s Rule
Find the Area of a Triangle on the Coordinate Plane Using a Determinant
Determine if a
Let
Determinant of matrix
Matrix
Submit Ourstion
Ask by Collins Ellis. in the United States
Mar 23,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
Matrix
Solution
- Write the matrix
:
- Compute the determinant of
by expanding along the first row:
-
Compute the
determinants: - For the first minor:
- The second term is multiplied by
, so it is . - For the third minor:
- For the first minor:
-
Substitute these results back into the expansion:
- Since
, the matrix is singular (noninvertible).
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Simplify this solution The Deep Dive
To find the determinant of the matrix
Calculating the minor determinants, we actually find that the determinant of
Interestingly, if a matrix has a determinant of zero, it implies that the matrix is singular, meaning it does not have an inverse. This can spur on interesting situations in linear algebra, especially when solving systems of equations where you might end up with dependent equations!
When applying determinants in real-world problems, you’ll find they can be crucial in fields such as physics and computer graphics. For example, calculating areas and volumes in multi-dimensional space often involves determinants. In graphics, they help determine transformations and affect object orientations, making sure things don’t get upside down or inside out!
