Matrices transform shapes and their determinant summarizes the effect on size and orientation. The absolute value indicates scale factor and sign describes flipping. Matrix multiplication determines transformation and the determinant’s product rule is useful. A matrix with determinant 0 cannot be canceled by another matrix, while a non-zero determinant matrix has an inverse.
Matrices are mathematical objects that transform shapes. The determinant of a square matrix A, denoted |A|, is a number that summarizes the effect A has on the size and orientation of a figure. If (ab) is the top row vector for A and (cd) is the bottom row vector, then |A| = ad-bc.
A determinant encodes useful information about how a matrix transforms regions. The absolute value of the determinant indicates the scale factor of the matrix, how much it stretches or shrinks a figure. Its sign describes whether the matrix flips the figures, producing a mirror image. Matrices can also tilt regions and rotate them, but this information is not provided by the determinant.
Arithmetically, the transforming action of a matrix is determined by matrix multiplication. If A is a 2 × 2 matrix with top row (ab) and bottom row (cd), then (1 0) * A = (ab) and (0 1) * A = (cd). This means that A brings the point (1,0) to the point (a,b) and the point (0,1) to the point (c,d). All matrices leave the origin unchanged, so we see that A transforms the triangle with endpoints at (0,0), (0,1) and (1,0) into another triangle with endpoints at (0,0), (a,b), and (c,d). The ratio of the area of this new triangle to that of the original triangle is equal to |ad-bc|, the absolute value of |A|.
The sign of the determinant of a matrix describes whether the matrix flips a shape. Considering the triangle with endpoints in (0,0), (0,1) and (1,0), if a matrix A keeps the point (0,1) stationary bringing the point (1,0) to the point (-1 ,0), so he flipped the triangle on the line x = 0. Since A flipped the figure, |A| it will be negative. The matrix does not change the size of a region, so |A| must be -1 to be consistent with the rule that the absolute value of |A| describes how much A stretches a figure.
Matrix arithmetic follows the associative law, which means that (v*A)*B = v*(A*B). Geometrically, this means that the combined action of first transforming a shape with matrix A and then transforming the shape with matrix B is equivalent to transforming the original shape with the product (A*B). It can be deduced from this observation that |A|*|B| = |A*B|.
The equation |A| * |B| = |A*B| has an important consequence when |A| = 0. In that case the action of A cannot be canceled by some other matrix B. This can be deduced by noting that if A and B were inverse, then (A*B) neither extends nor flips any region, so |A *B| = 1. Since |A| * |B| = |A*B|, this last observation leads to the impossible equation 0 * |B| = 1.
The converse statement can also be proved: if A is a square matrix with non-zero determinant, then A has an inverse. Geometrically, this is the action of any matrix that does not flatten a region. For example, squashing a square into a line segment can be undone by another matrix, called an inverse. This inverse is the matrix analog of a reciprocal.
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