Question

Let $A,BandC$ be $2\times 2$ matrices with entries from the set of real numbers. Define $\ast$ as follows:
$A\ast B=\frac{1}{2}(AB{}^{\prime }+A^{\prime }B)$. Which of the given is true?

• 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. $A\ast I=A+A^{\prime }$

Answer 1

Answer: (A), (C)

The correct options are
A $A\ast B=B\ast A$
C $A\ast (B+C)=A\ast B+A\ast C$

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

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

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

3)$A\ast (B+C)=\frac{1}{2}\left(A\right(B+C{)}^{\prime }+A^{\prime }(B+C))$

$=\frac{1}{2}(A{B}^{\prime }+A^{\prime }B)+\frac{1}{2}(A{C}^{\prime }+A^{\prime }C)$

$=A\ast B+A\ast C$

4)$A\ast I=\frac{1}{2}(A{I}^{\prime }+A^{\prime }I)=\frac{1}{2}(A+A^{\prime })$

Hence, options A and C.