matrix divison is possible?

Question

i have a question
we all know,
A inverse = adj. A/det. A
so, we can write
det. A = adj. A/A invese
since, adj A & A inverse both are matrices hence result of matrix divison is always a scaler unit


but all teachers says its wrong but no one could give satisfactory answer
can anyone help me ???????

Answer 1

This is incorrect. The expression 1/A^(-1) is meaningless - you have utilized a scalar operation on a matrix quantity without justification. What does it mean to take the reciprocal of a matrix? Here is your task. Consider the 2 by 2 identity matrix. What is the value of its reciprocal?



Edit 1: Also, your expression of det(A) = adj(A)/A^(-1) gives no information about how to divide an arbitrary matrix A by another matrix B. What does A/B mean?



Edit 2: More importantly - all operations are defined. For the operation of "matrix addition" we use the term addition and the notation + because the matrix operation A+B is useful and shares analogy with the already prior defined addition of real numbers. To say that matrix division does not exist is just saying no mathematician has defined a consistent operation that we have called division. For example, I could define the operation "/" as A/B=2 for all matrices A and B, but this would be a useless operation. So, basically, mathematicians have not found a useful, consistent operation that is somewhat analogous to division that we would give meaning to A/B.

Answer 2

some notation may look like division but this is really multiplication by the inverse matrix. Only applies to square matrices. The result of multiplying matrices is a matrix.

Matlab has this notation \. The APL programming language uses "domino".

Answer 3

and what is the matrix A divided by the matrix B ..A is [ 4 x 1 ] , B = [ 7 x 3 ] ?...that is why matrix division is not defined