Determinant - best way of introducing determinants on a linear algebra course

In summary, the determinant is a valuable tool for solving equations and understanding the geometry of space. It is important to stress the importance of the determinant in introductory linear algebra courses.
  • #1
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Determinant -- best way of introducing determinants on a linear algebra course

What is the best way of introducing determinants on a linear algebra course? I want to give real life examples of where the determinant is applied.
 
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  • #2


Something you absolutely need to do is mention that the determinant is the signed volume of a parallelepiped. That demystifies a lot of properties of the determinants.

Also, be sure to mention the history of the determinant: that it was invented to see when systems of equations have a solution and when not. Showing explicitely that the system

[tex]\left\{\begin{array}{ll} ax+by=0\\ cx+dy=0\end{array}\right.[/tex]

has a unique solution if and only if ad-bc=0 is very illuminating. It makes clear why the determinant was invented in the first place: to generalize this to higher dimensions.
 
  • #3


If you have a square matrix, then its determinant is equal to the volume of the solid parellelepiped spanned by the column vectors (or the row vectors). That is the most concrete application of the value of the determinant that I know of. Actually, the determinant gives a signed volume which is positive or negative depending on the orientation of the vectors.

Most of the applications in algebra only test whether or not the determinant is 0 or not zero. A nonzero determinant means you can solve some equations as micromass said. But in calculus the actual value matters. For example the change of variables formula for integrals of multiple variables uses determinants in this way. (This would be hard to explain to students who don't know what the Jacobian is)

To prove that it is equal to the volume, you need to show that the volume is alternating, multilinear, and equals 1 for the unit cube (the properties that uniquely characterize the determinant). Obviously the volume of the unit cube is 1. The other two properties are essentially equivalent to Vol= Base*height. Alternatively, in 2 dimensions, you can prove the area formula just by drawing a picture and working it out explicitly.
 
  • #4


another way to think about it is to convince yourself that in the n^2 dimensional space of all nbyn matrices, the singular ones have codimenson one, so there should be an equations that vanishes exactly on those.
 
  • #5


One effective way to introduce determinants in a linear algebra course is to first provide a clear definition of what a determinant is and its mathematical representation. This can be followed by demonstrating how determinants are used to solve systems of linear equations and finding the inverse of a matrix.

To give real-life examples, you can discuss how determinants are used in engineering and physics to solve problems related to forces, moments, and equilibrium. For instance, in structural engineering, determinants are used to determine the stability of a structure by analyzing the forces acting on it. In physics, determinants are used to calculate the angular momentum of a rotating object.

In addition, you can also discuss how determinants are used in computer graphics to transform and rotate objects in 3D space. This can help students see the practical applications of determinants in a visual and interactive way.

Another approach could be to introduce real-world scenarios where determinants are used to solve problems, such as in economics to analyze supply and demand curves or in genetics to study inheritance patterns.

Overall, incorporating real-life examples and applications of determinants can help students see the relevance and importance of this mathematical concept in various fields and make it more engaging and relatable for them.
 

1. What is a determinant?

A determinant is a numerical value that can be calculated from a square matrix. It is often used in linear algebra to help solve systems of equations and determine properties of matrices.

2. Why is the determinant important in linear algebra?

The determinant is important because it can tell us if a matrix is invertible (has an inverse) and if the system of equations represented by the matrix has a unique solution. It is also used to calculate the area or volume of a transformation in multi-dimensional space.

3. How do you calculate the determinant of a matrix?

There are a few different methods for calculating determinants, such as using cofactor expansion or using row operations. However, the most efficient and commonly used method is Gaussian elimination, which involves using elementary row operations to transform the matrix into an upper triangular form and then multiplying the diagonal entries to find the determinant.

4. Can the determinant be negative?

Yes, the determinant can be negative. It is a numerical value and can be positive, negative, or zero depending on the matrix. In fact, the sign of the determinant can tell us about the orientation of the transformation represented by the matrix.

5. What is the relationship between determinants and eigenvalues?

Eigenvalues are closely related to determinants. The eigenvalues of a matrix are the solutions to the characteristic equation, which involves the determinant of the matrix. Additionally, the determinant can also be used to calculate the trace (sum of eigenvalues) of a matrix.

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