#) Determinants: In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well.

{+} The determinant provides important information 1) about a matrix of coefficients of a system of linear equations.

{+} about a matrix that corresponds to a linear transformation of a vector space.

{+} In the first case the system has a unique solution exactly when the determinant is nonzero; when the determinant is zero there are either no solutions or many solutions. In the second case the transformation has an inverse operation exactly when the determinant is nonzero.

{+} A geometric interpretation can be given to the value of the determinant of a square matrix with real entries: the absolute value of the determinant gives the scale factor by which area or volume (or a higher-dimensional analogue) is multiplied under the associated linear transformation, while its sign indicates whether the transformation preserves orientation.

{+} Thus a 2 × 2 matrix with determinant −2, when applied to a region of the plane with finite area, will transform that region into one with twice the area, while reversing its orientation.


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