Question

given three conformable matrices, a, b, and c, which of the following rules is correct?

a. none of the other answers is correct.

b. ( ab + bc = b(a + c) )

c. ( a(b + c) = (b + c)a )

d. ( abc = acb )

Explanation:

Step1: Recall Matrix Multiplication Properties

Matrix multiplication is not commutative in general, meaning \( XY
eq YX \) for matrices \( X,Y \) (unless special cases like diagonal matrices with same diagonal elements, etc.). Also, distributive property: \( A(B + C)=AB + AC \), and factoring: \( AB+BC \) – check if \( B(A + C)=BA + BC \), but \( AB
eq BA \) usually.

Step2: Analyze Option b

Check \( AB + BC \) vs \( B(A + C) \). By distributive law, \( B(A + C)=BA + BC \). Since matrix multiplication is non - commutative, \( AB
eq BA \) (in general), so \( AB + BC
eq BA + BC = B(A + C) \) (unless \( AB = BA \), but not general).

Step3: Analyze Option c

\( A(B + C)=AB + AC \) (distributive). \( (B + C)A=BA + CA \). Since \( AB
eq BA \) and \( AC
eq CA \) (in general), \( A(B + C)
eq(B + C)A \).

Step4: Analyze Option d

\( ABC = A(BC) \), \( ACB=A(CB) \). Since \( BC
eq CB \) (in general), \( A(BC)
eq A(CB) \), so \( ABC
eq ACB \).

Step5: Conclusion

Since options b, c, d are incorrect (in general, for conformable matrices, without special conditions), option a is correct.

Answer:

a. None of the other answers is correct.