Understanding Determinants: Proof of det(kA) = k^ndetA and det(rI(n)) = r^(n)

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In summary, the conversation discusses the proof of det(kA) = k^ndetA and the confusion around det(rI(n)) = r^(n) and det(rA) = r^ndetA, with the clarification that I is the identity matrix and det(I) is always 1.
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captainjack2000
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1. The proof that det(kA) = k^ndetA where A is nxn
I read somewhere that det(rI(n)) = r^(n)
so det(rA) = det(rI(n).A) = r^ndetA but I am really confused about how they got that? Is I the identity matrix? What would the det(I) be?
 
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captainjack2000 said:
1. The proof that det(kA) = k^ndetA where A is nxn
I read somewhere that det(rI(n)) = r^(n)
so det(rA) = det(rI(n).A) = r^ndetA but I am really confused about how they got that? Is I the identity matrix? What would the det(I) be?


You pretty much got it.

We know that det(AB) = det(A)det(B).

Therefore,

det(rA) = det(rI[n]A) = det(rI[n])det(A) = r^n det(I[n]) det(A) = r^n (1) det(A) = r^n det(A)

I showed every step possible basically.

Yes, the determinant of I[n] is always 1. It is the identity matrix.
 

What is a determinant and why is it important?

A determinant is a mathematical value that is calculated for a square matrix. It is important because it helps determine if a matrix is invertible, which is crucial in solving systems of equations and in other mathematical operations.

How do I calculate the determinant of a matrix?

To calculate the determinant of a matrix, you can use the Laplace expansion method or the Gaussian elimination method. Both methods involve manipulating the matrix through a series of operations to get it into a triangular form, then multiplying the values on the diagonal to get the determinant.

Can determinants be negative?

Yes, determinants can be negative. The sign of a determinant is determined by the number of row swaps needed to get the matrix into triangular form. An odd number of swaps will result in a negative determinant, while an even number will result in a positive determinant.

What is the difference between a determinant and an eigenvalue?

A determinant is a value calculated for a matrix, while an eigenvalue is a value associated with a specific vector in that matrix. The determinant helps determine if a matrix is invertible, while the eigenvalue helps determine the direction and magnitude of the transformation performed by the matrix on that vector.

How are determinants used in real life?

Determinants have many practical applications in fields such as engineering, physics, and economics. They are used to solve systems of linear equations, calculate volumes and areas, and determine the stability of structures and systems. They are also used in computer graphics and machine learning algorithms.

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