SOLUTION: I want to know what exactly is a determinant? Finding the determinant and solving for it is something I know very well. But why do we solve it And what does the solution actual

Question 763380: I want to know what exactly is a determinant?
Finding the determinant and solving for it is something I know very well.
But why do we solve it
And what does the solution actually determine?
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
Welcome to multi-variable calculus!
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The determinant can be thought of as a function whose input is a square matrix and whose output is a number.
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A 2 by 2 determinant is used to calculate the area of a parallelogram and a 3 by 3 determinant is used to calculate the volume of parallelepiped.
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Determinants of n by n matrices and their associated linear transformations provide us with important geometric properties. I enjoyed my college course, "Calculus on Manifolds" which was a graduate course I took as a senior.
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An example, one-dimensional linear transformation is the function T(x)=3x. We view it as a mapping from the real line R back onto another the real line R'. Let's consider the domain as the closed interval [0,1] and we see that the range is [0,3]. The fact that the determinant of the matrix associated with T is 3 means that T stretches objects so that their length is increased by a factor of 3.
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This gives you a flavor of what is ahead if you take advanced mathematics - lots of FUN!
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