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If A = [{:(1,2),(-2,3):}], B= [{:(2,1),(...

If `A = [{:(1,2),(-2,3):}], B= [{:(2,1),(2,3):}] and C = [{:(-3,1),(2,0):}]`,verify that (AB)C = A(BC) and A(B+C)=AB+AC.

Text Solution

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we have, AB `AB = [{:(1,2),(-2,3):}],xx [{:(2,1),(2,3):}]`
`=[{:(1.2+2.2,1.1+2.3),((-2).2+3.2,(-2).1+3.3):}]=[{:(6,7),(2,7):}]`
`BC =[{:(2,1),(2,3):}]xx[{:(-3,1),(2,0):}]=[{:(2*(-3)+1.2,2.1+1.0),(2*(-3)+3.2,2.1+3.0):}]`
`=[{:(-6+2,2+0),(-6+6,2+0):}]=[{:(-4,2),(0,2):}]`
`BC =[{:(2,1),(-2,3):}]xx[{:(-3,1),(2,0):}]=[{:(1*(-3)+2.2,,1.1+2.0),((-2)*(-3)+3.2,,(-2)1+3.0):}]`
`=[{:(-3+4,1+0),(6+6,-2+0):}]=[{:(1,1),(12,-2):}]` ltbegt `B+C =[{:(2,1),(2,3):}]+[{:(-3,1),(2,0):}]=[{:(2-3,1+1),(2+2,3+0):}]=[{:(-1,2),(4,3):}]`
Now, (AB)C `[{:(6,7),(2,3):}]xx[{:(-3,1),(2,0):}]=[{:(-18-3,6+0),(-6+14,2+0):}]`
`=[{:(-4,6),(8,2):}]`
A(BC)` =[{:(1,2),(-2,3):}]xx[{:(-4,2),(0,2):}]`
`=[{:(-4+0,2+4),(8+0,-4+6):}]=[{:(-4,2),(8,2):}]`
Thus, from Eqs. (i) and (ii), we get, (AB)C =A(BC)
Now, A(B+C) `=[{:(1,2),(-2,3):}]xx[{:(-1,2),(4,3):}]=[{:(-1+8,2+6),(2+12,-4+9):}]`
`=[{:(7,8),(14,5):}]`and AB+AC `=[{:(6,7),(2,7):}]xx[{:(1,1),(12,-2):}]=[{:(6+1,7+1),(2+12,7-2):}]`
`=[{:(7,8),(14,5):}]`
Thus, from Eqs, (iii) and (iv), we get
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