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One of the reasons for the importance of the matrix exponential is that it can be used to solve systems of linear ordinary differential equations.The solution of = (), =, where A is a constant matrix and y is a column vector, is given by =.
Consider the system of equations + + = + + = + + = The coefficient matrix is = [], and the augmented matrix is (|) = []. Since both of these have the same rank, namely 2, there exists at least one solution; and since their rank is less than the number of unknowns, the latter being 3, there are an infinite number of solutions.
Familiar properties of numbers extend to these operations on matrices: for example, addition is commutative, that is, the matrix sum does not depend on the order of the summands: A + B = B + A. [9] The transpose is compatible with addition and scalar multiplication, as expressed by (cA) T = c(A T) and (A + B) T = A T + B T. Finally, (A T) T = A.
The sum and difference of two symmetric matrices is symmetric. This is not always true for the product: given symmetric matrices and , then is symmetric if and only if and commute, i.e., if =.
In linear algebra, an invertible matrix is a square matrix that has an inverse.In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by an inverse to undo the operation.
GE multifactorial analysis is a technique used in brand marketing and product management to help a company decide what products to add to its portfolio and which opportunities in the market they should continue to invest in.