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I understand how it is calculated, but what does the determinant of a matrix mean and what is it used for? If you could explain it to me in simple words, thank you.

zibzabpuzzless
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    This is almost like asking "what is the real meaning of the number $e$?" I don't say that to be critical, but to point out that there is no simple answer. The determinant has many important properties that can be considered part of its meaning, and choosing one such property as the "real" meaning will lead to a narrow understanding of the subject. – jack Mar 30 '15 at 06:10

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The absolute value of the determinant of a matrix is the volume of the parallepiped spanned by the vectors in the matrix.

Parallelpiped is the 3d version of a parallelogram. The rows of the matrix are called vectors. These vectors make up the sides for the parallelpiped.

Cameron
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The determinant of a matrix is defined as a value associated with square matrix.

Some applications, but not limited to:

  • Particularly a useful tool when solving linear equations. One example is to determine whether the linear system has an unique solution.
  • Finding the inverse of a matrix
  • Some calculus applications (e.g. Jacobian Determinant)
Joshua
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It's a function designed to take in a square matrix and output a number that tells us the linear dependence of vectors that make up the matrix.

This problem can be interpreted many contexts and so the determinant can be viewed in different ways.

One such applications is the area of a parallelogram defined by two vectors ($2\times2$ case). Herbert Gross Mit lectures go into more detail and are an excellent resource.

Karl
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