Determinant Expansions: Intuitive Understanding & Proofs

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In summary, the determinant expansion is the same regardless of which row or column is chosen, as it is a property of the map represented by the matrix and not the details of the matrix itself. The proof for this is too technical to explain, but it can be found online.
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heshbon
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Hi, I can't understand why any determinant expansion is the same via any row/column.
My lecturer says the proof is too technical to go over,
Does anyone have a good way to think about it intuitavely or know a site which has a full proof?
thnx
 
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This is a bit of a vague argument, but I think it is the underlying thought:
The determinant is not so much a property of a matrix, but actually a property of the map which that matrix represents (and usually as such can be assigned a geometric meaning). Therefore, it should not depend on details like which basis we happen to write the matrix in.

I'm not sure we can offer you much rigor without actually going into the proof :smile:

I think the proof is here by the way, at least for expansion along the first row and column. You can just do a basis transformation (permute the basis vectors) to get any row/column you want as the first one.
 
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FAQ: Determinant Expansions: Intuitive Understanding & Proofs

What is a determinant expansion?

A determinant expansion is a mathematical process used to find the value of a determinant, which is a special number associated with a square matrix. It involves breaking down the matrix into smaller submatrices and using certain formulas to calculate the determinant of each submatrix, then combining these values to find the overall determinant of the original matrix.

Why is it important to understand determinants?

Determinants have many important applications in mathematics, physics, and engineering. They are used to solve systems of linear equations, find inverse matrices, and calculate the areas and volumes of geometric shapes. Understanding determinant expansions can help improve problem-solving skills and build a strong foundation for more advanced mathematical concepts.

Can you provide an intuitive explanation of determinant expansions?

Think of a determinant as a "size" or "scaling factor" of a matrix. When we expand a determinant, we are essentially breaking it down into smaller pieces and finding the scaling factor for each piece. Then, by combining these scaling factors, we can determine the overall scaling factor for the original matrix.

Are there different methods for expanding determinants?

Yes, there are several methods for expanding determinants, including cofactor expansion, row reduction, and using properties of determinants. Each method may be more suitable for certain types of matrices or situations, so it is important to understand multiple approaches.

How can I prove the validity of a determinant expansion?

The validity of a determinant expansion can be proven using mathematical induction, which involves showing that the expansion holds true for a base case (e.g. a 2x2 matrix) and then showing that it also holds true for a larger case (e.g. a 3x3 matrix) by using the same logic and formulas. Alternatively, one can also prove the validity of a determinant expansion by using the properties of determinants and matrix operations.

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