Why is (AB)^-1 = B^-1A^-1 in matrices, and not (AB)^-1 = A^-1B^-1
let A be an m by n matrix and B be a n by p matrix. Then A*B is defined. Also, A^-1 is an n by m matrix, and B^-1 is an p by n matrix. So B^-1 * A^-1 is also defined, but A^-1 * B^-1 may not be.
Or, in other words, the commutative property of scalar multiplication does not apply to matrices.
To understand why (AB)^-1 equals B^-1A^-1 in matrices, let's break down the process of finding the inverse of a matrix.
In a matrix equation, AB represents the product of matrices A and B. To solve for the inverse of this product, (AB)^-1, we need to find a matrix that, when multiplied by AB, yields the identity matrix (I).
To begin, let's assume that the matrices A and B have inverses, denoted as A^-1 and B^-1, respectively. These inverses exist when the original matrices are non-singular (i.e., their determinants are non-zero).
Firstly, we can multiply both sides of the equation AB = I by A^-1 on the left to isolate matrix B:
A^-1(AB) = A^-1I
By the associative property, we can rewrite the left-hand side as:
(A^-1A)B = A^-1
Since A^-1A results in the identity matrix (AA^-1 = I), we have:
IB = A^-1
Thus, we've obtained the equation B = A^-1.
Next, let's switch the equation around to obtain B^-1 on the left side:
B^-1 = (A^-1)B
Finally, if we compare this equation with the original goal of finding (AB)^-1, we can conclude that (AB)^-1 equals B^-1A^-1.
So, (AB)^-1 = B^-1A^-1.
In summary, by multiplying the inverses of matrix B and matrix A in reverse order, we can find the inverse of their product (AB). However, it is crucial to note that this equation holds true only when both matrices A and B are non-singular (i.e., their determinants are non-zero).