Thank you Sir for your valuable paper. In Section II Matrix Division, we will check "The Division Algorithm" for matrix multiplication. This way will be solving our problems.
Thank you very much for your contribution to science. This has been done in my recent study. My study is under review. Preparing for publication. This subject is still a fresh and untouched field, we can also study together. Thank you again for your kind thought.
Rational matrices, like rational numbers, can be simplified and expanded using similar principles. A rational matrix is a matrix whose elements are rational numbers (fractions). Simplifying and expanding rational matrices involve performing operations analogous to those with rational numbers, such as reducing fractions and finding common denominators.
Simplifying Rational Matrices:To simplify a rational matrix, you aim to reduce its elements to their simplest form. This involves dividing both the numerator and denominator of each element by their greatest common divisor (GCD). For example: Consider the matrix A with rational elements: A=[[2/4,3/6],[5/10,4/8]]. To simplify A, divide all elements by their respective GCDs: A=[[1/2,1/2],[1/2,1/2]]
Expanding Rational Matrices:Expanding a rational matrix is not as straightforward as with scalar numbers, but you can apply similar concepts when performing matrix operations. For example: Consider the matrix B with rational elements and a scalar factor of 2: B=[[2/3,1/4],[3/5,1/6]]. To expand B by a scalar factor of two, you would multiply each element by two: 2B=[4/3,1/2],[6/5,1/3]]. Expanding by scalar multiplication is a common operation, but note that other types of matrix operations, such as matrix addition and multiplication, involve more complex calculations.
Keep in mind that not all operations that are valid for scalar rational numbers have direct analogs for matrices. Division, for example, is not directly defined for matrices in the same way it is for scalar numbers. You might use techniques like matrix inversion instead of division when dealing with matrices.
Additionally, some operations in matrix algebra, like matrix inversion, determinant calculation, and matrix multiplication, can lead to rational numbers becoming irrational or even undefined if not handled properly. Always ensure you're following established rules and guidelines for matrix operations to maintain accuracy and validity.
Certainly, rational matrices can undergo simplification and expansion, akin to rational expressions. A rational matrix comprises entries represented as rational functions, which are ratios of polynomials. Similar to algebraic fractions, rational matrices can be simplified by canceling common factors in the numerators and denominators of the entries. Likewise, expanding rational matrices involves obtaining a common denominator and performing arithmetic operations on the rational expressions within the entries. It's essential to exercise caution to ensure that operations are carried out properly, as rational matrix operations can become intricate due to matrix dimensions and algebraic complexity. Diligence in computation is pivotal to ensure the validity and accuracy of the resultant matrices.