November 25, 2024
The fundamental property of a matrix is the transformation space, so eigenvalues describe the scaling strength in a certain direction. Eigenvectors describe that direction.
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November 21, 2024
The determinant has nothing to do with “equations”, it is a scalar.
The determinant is only applicable to square matrices, describing the volume of n n-dimensional vectors. For two dimensions, it is the area. A determinant of zero indicates that the matrix cannot span an n-dimensional space, at most n-1 dimensions, i.e., not full rank. So the volume is zero in n dimensions.
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March 18, 2024
Matrix multiplication has four cases, but essentially they are all vector operations.