Respuesta :
Answer:
The formula is not valid because the commutative property with respect to the matrix product operation is not fulfilled in the vector space of the real matrices.
Step-by-step explanation:
The formula is not valid because the commutative property with respect to the matrix product operation is not fulfilled in the vector space of the real matrices. That is, AB does not necessarily equal BA.
[tex](A+B)(A-B) = A^2-AB+BA-B^2\neq A^2 - B^2[/tex]
[tex]A=\left[\begin{array}{ccc}1&0&0\\0&0&6\\0&8&0\end{array}\right] \\B=\left[\begin{array}{ccc}0&2&0\\6&0&0\\0&0&9\end{array}\right] \\(A -B) = \left[\begin{array}{ccc}1&-2&0\\-6&0&6\\0&8&-9\end{array}\right]\\\\(A + B) = \left[\begin{array}{ccc}1&2&0\\6&0&6\\0&8&9\end{array}\right]\\(A - B)(A + B) = \left[\begin{array}{ccc}-11&2&-12\\-6&36&54\\48&-72&-33\end{array}\right]\\A^2 - B^2 = \left[\begin{array}{ccc}-11&0&0\\0&36&0\\0&0&-33\end{array}\right]\\[/tex]
You can use the fact that multiplication of matrices is dependent on the order of the matrices which are multiplied.
The correct option for the given condition is
Option C: The formula is not valid because in general, AB ≠ BA for matrices.
Why is it that AB ≠ BA for two matrices A and B usually?
It might be that AB = BA for two matrices A and B but it is very rare and thus, cannot be generalized as identity.
Suppose A has got shape (m,n) (m rows, n columns)
and B has got shape (n,k) (n rows, k columns), then AB is defined but BA is not defined if k ≠ m.
Also, even if k =m, we can't say for sure that AB = BA
Thus, usually we have AB ≠ BA
Using the above fact to and distributive property to evaluate (A + B)(A − B)
For two matrices A and B, supposing that AB and BA are defined, then we have
[tex](A+B)(A-B) = A(A-B) + B(A -B) = A^2 -AB + BA - B^2[/tex]
Since may or may not have AB equal to BA, thus, we cannot cancel those two middle terms to make 0 matrix.
Thus,
The correct option for the given condition is
Option C: The formula is not valid because in general, AB ≠ BA for matrices.
Learn more about matrix multiplication here:
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