Determinant

In linear algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. Determinants are important both in calculus, where they enter the substitution rule for several variables, and in multilinear algebra.

Determinants of 2-by-2 matrices

The 2×2 matrix

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:A=egin{bmatrix}a&b\

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c&dend{bmatrix}

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has determinant

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:det(A)=ad-bc ,.

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The interpretation when the matrix has real number entries is that this gives the area of the parallelogram with vertices at (0,0), (a,c), (b,d), and (a + b, c + d), with a sign factor (which is −1 if A as a transformation matrix flips the unit square over).

Related Topics:
Area - Parallelogram - Transformation matrix - Unit square

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A formula for larger matrices will be given below.

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