How can I divide the two matrices together ?

Hint: First of all , we need to know that the matrix is a rectangular arrangement of elements ( numbers ) into the rows and columns.Technically , the matrices cannot be divided . But if you remember the basic concept of division of two fractions , then one fraction remains the same but the other fraction gets inverted and multiplied to the former fraction for performing the division . In the similar way , if we have to divide two matrices together we must take the inverse of one matrix and multiply it with the other matrix .

Complete answer:
So if we have to divide two matrices together we must take the inverse of one matrix and multiply it with the other matrix . But for taking out the matrix we must check the divisor matrix is square otherwise there will be no unique solution.

After checking for invertible property , we must check for the multiplication of two matrices that can happen or not. For that we need to check the number of columns in the first matrix must be the same as the number of rows in the second matrix .

Furthermore , Find the determinant of the matrix . if the determinant is non zero we will be able to find the inverse of that matrix . Now we will invert the matrix and then take the reciprocal of the determinant and multiply them together. We get a new matrix that is the final resulting inverse of the matrix. After this , multiply the inverse of one matrix by another original matrix , then we will get the required solution to the problem .

Note:The division of two matrices is an undefined function instead we solve \[\left[ A \right]{\text{ }} \times {\text{ }}{\left[ B \right]^{ - 1}}\] .Always remember that \[\left[ A \right]{\text{ }} \times {\text{ }}{\left[ B \right]^{ - 1}}\] may have a different answer than calculating for \[{\text{ }}{\left[ B \right]^{ - 1}} \times \left[ A \right]{\text{ }}\]. These are two different problems that can have different solutions .

Keep in mind that if you want to take out the inverse of a matrix you must have a square matrix having equal number of rows and columns otherwise there will be no unique solution of the matrices . Must check the number of columns in the first matrix must be the same as the number of rows in the second matrix for the multiplication of two matrices otherwise there is no solution .

The number of rows and columns are the same for the original matrix \[\;\left[ B \right]\] as that of inverse of that same matrix \[\;{\left[ B \right]^{ - 1}}\;\].If the determinant of the matrix comes out to be 0 then the inverse of the matrix does not exist.If we multiply the inverse by the original matrix then their product will be equal to the identity matrix always .
Matrix multiplication is not commutative .