How do you multiply $A = ((1,0,2),(3,-1,0),(0,5,1))$ with $B = ((1,1,0),(0,2,1),(3,-1,0))$ ?

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Answer 1 · 2016-09-17T17:26:05

$[AB]=[(7,-1,0),(3,1,-1),(3,9,5)]$ .

$[BA]=[(4,-1,2),(6,3,1),(0,1,6)]$ .

The Multiplication can occur in two ways $(1) :[AB, and, (2) : BA$ .

In General, $ABneBA$ .

Let $A_(3xx3)=[a_(ij)]$ and $B_(3xx3)=[b_(jk)]$

For $AB$ to be defined, we must have,

No. of Columns in $A$ =no. of Rows in $B$ .

We find that this cond. is satisfied in our case, so, $AB$ is defined.

Let $AB=C=[c_(ik)]_(3xx3),$ where,

$c_(ik)=sum_(j=1)^(j=3) (a_(ij)*b_(jk)), i,k=1,2,3$ .

$:. c_11=sum_(j=1)^(j=3) (a_(1j)*b_(j1))=a_11*b_11+a_12*b_21+a_13*b_31$

$:. c_11=1*1+0*0+2*3=7$ . Similarly,
$c_12=1*1+0*2+2(-1)=-1$
$c_13=1*0+0*1+2*0=0$

$c_21=3*1+(-1)0+0*3=3$
$c_22=3*1+(-1)2+0*(-1)=1$
$c_23=3*0+(-1)1+0*0=-1$

$c_31=0*1+5*0+1*3=3$
$c_32=0*1+5*2+1*(-1)=9$
$c_33=0*0+5*1+1*0=5$

$:. [C]=[AB]=[(7,-1,0),(3,1,-1),(3,9,5)]$ .

Similarly, we can work out, $[BA]=[(4,-1,2),(6,3,1),(0,1,6)]$ .

Enjoy Maths.!

How do you multiply A = ((1,0,2),(3,-1,0),(0,5,1))A = ((1,0,2),(3,-1,0),(0,5,1)) with B = ((1,1,0),(0,2,1),(3,-1,0))B = ((1,1,0),(0,2,1),(3,-1,0))?