The determinant of a square matrix A denoted by det(a), is a scalar that measures the change in “magnitude” or “size” of the linear transformation denoted by a matrix. For example, if a linear transformation increases the size of a square represented by the coordinates of the vectors (0, 1) and (1,0), and after a matrix transformation it becomes a rectangle of height 2 and width 3, the determinant of such a linear transformation is 6, since it’s area was increased by 2 x 3 = 6.
This also means that if the determinant of a matrix is 0, it has transformed the area taken up by our coordinates to 0. In other words, if the determinant of a matrix is 0, it squishes the area taken up by any geometry in that space to a smaller dimension. For example, if in 3d space, a certain transformation has a matrix with a determinant of zero, it would squish all geometry, i.e. the volume of any geometry in 3d space to 0, either by taking it to a plane, a line, or a point.
In a square matrix of the form
Row operations are linear transformations. Each corresponds to a matrix called an elementary matrix.
The determinant of the inverse of a matrix is 1/det(A).
The determinant of a matrix times a certain scalar alpha is equal to the scalar to the power of the n rows (or columns in a square matrix) such as:
The determinant of a matrix that has multiple components is the same as the addition of the two determinants separately? So that
det(a + b) = det(a)l + det(b)?
Properties
A matrix with a determinant of zero indicates linear dependence of the row space of that matrix.
A matrix with a non-zero determinant also indicates that it is invertible.