Question

Let $A,BandC$ be $2\times 2$ matrices with entries from the set of real numbers. Define $\ast$ as follows: $A\ast B=\frac{1}{2}(AB+BA)$, then

• A. $A\ast B=B\ast A$
• B. $A\ast A=A^2$
• C. $A\ast (B+C)=A\ast B+A\ast C$
• D. $A\ast I=A$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $A\ast B=B\ast A$
B $A\ast A=A^2$
C $A\ast (B+C)=A\ast B+A\ast C$
D $A\ast I=A$

$A\ast 1=\frac{1}{2}(AI+IA)=\frac{1}{2}(A+A)=A$

$\therefore A\ast I=A,\therefore$ (a)holds.

$A\ast A=\frac{1}{2}(AA+AA)=\frac{1}{2}(A^2+A^2)=A^2$

$\therefore$ (b) holds.

$\left(c\right)A\ast B=\frac{1}{2}(AB+BA)B\ast A=\frac{1}{2}(BA+AB)=\frac{1}{2}(AB+BA)$

$[\because$ addition is commutative$]$

$\therefore A\ast B=B\ast A,\therefore$ (c)holds

$A\ast (B+C)=\frac{1}{2}(A(B+C)+(B+C)A)=\frac{1}{2}(AB+CA+BA+CA)$

$=\frac{1}{2}(AB+BA)+\frac{1}{2}(AC+CA)=A\ast B+A\ast C$

$\therefore$ (d) holds.