Single Matrix
The determinant of a square matrix is a number that indicates how that matrix behaves. There are different ways to calculate the determinant depending on the size of the individual matrix. The simplest form concenrs a 2x2 matrix:
The determinant can be negative or zero. A negative determinant means that the volume was mirrored over an odd number of axes. If the determinant is zero, then there is either no unique non-trivial solution, or there are infinitely many.
For a 3x3 matrix, you must multiply the column in a specific sequence. This formula is known as the "Rule of Sarrus":
Alternatively, this same calculation for a 3x3 dimensional matrix can also be mathematically perceived as follows:
Multiplying Matrices
Next, there are some arithmetical operations which can be performed on multiple matrices. The simplest scenario is when both matrices are symmetrical. Otherwise there are some rules first which must be applied depending on the dimensions of these unsymmetrical matrices:
- The multiplication of matrices is not commutative ⇒ A⋅B ≠ B⋅A
- The overriding rule is that A can only be multiplied by B if the number of columns in A matches the number of rows in B.
- Hence, if A which is a mxn dimensional matrix is multiplied by B which is a nxp dimensional matrix then the resulting product C will be a matrix with mxp dimensions.
- The product of B⋅A can not be evaluated {or is invalid} if the number of columns in B does not match the number of rows in A.
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