Question

Explain why the formula is not valid for matrices. Illustrate your argument with examples. (A + B)(A − B) = A2 − B2 The formula is not valid because in general, the distributive property is not valid for matrices. The formula is not valid because in general, B(−B) ≠ −B2 for matrices. The formula is not valid because in general, AB ≠ BA for matrices. The formula is not valid because in general, A(−B) ≠ −AB for matrices. Select the pair of matrices, A and B, for which the formula is not valid.

Answer 1

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.

$(A+B)(A-B) = A^2-AB+BA-B^2\neq A^2 - B^2$

$A=\left[\begin{array}{ccc}1&0&0\\0&0&6\\0&8&0\end{array}\right] \\B=\left[\begin{array}{ccc}0&2&0\\6&0&0\\0&0&9\end{array}\right] \\(A -B) = \left[\begin{array}{ccc}1&-2&0\\-6&0&6\\0&8&-9\end{array}\right]\\\\(A + B) = \left[\begin{array}{ccc}1&2&0\\6&0&6\\0&8&9\end{array}\right]\\(A - B)(A + B) = \left[\begin{array}{ccc}-11&2&-12\\-6&36&54\\48&-72&-33\end{array}\right]\\A^2 - B^2 = \left[\begin{array}{ccc}-11&0&0\\0&36&0\\0&0&-33\end{array}\right]$