निम्नलिखित आव्यूहों को एक सममित आव्यूह तथा एक विषम सममित आव्यूह के योगफल के रूप में व्यक्त कीजिए :
(i) `[{:(3,5),(1,-1):}]`
(ii) `[{:(6,-2," "2),(-2," "3,-1),(2,-1," "3):}]`
(iii) `[{:(3,3,-1),(-2,-2," "1),(-4,-5," "2):}]`
(iv) `[{:(1,5),(-1,2):}]`

Explanation

(i) माना `A=[{:(3,5),(1,-1):}]`
`rArrA'=[{:(3,5),(1,-1):}]=[{:(3,1),(5,-1):}]`
`thereforeA+A'=[{:(3,5),(1,-1):}]+[{:(3,1),(5,-1):}]=[{:(6,6),(6,-2):}]`
`rArr(1)/(2)(A+A')=[{:(3,3),(3,-1):}]`
तथा `A-A'=[{:(3,5),(1,-1):}]-[{:(3,1),(5,-1):}]=[{:(0,4),(-4,0):}]`
`rArr(1)/(2)(A-A')=[{:(0,2),(-2,0):}]`
अब, `A=(1)/(2)(A+A')+(1)/(2)(A-A')`
`rArr[{:(3,5),(1,-1):}]=[{:(3,3),(3,-1):}]+[{:(0,2),(-2,0):}]`
(ii) माना `A=[{:(6,-2," "2),(-2," "3,-1),(2,-1," "3):}]`
`A=[{:(6,-2," "2),(-2," "3,-1),(2,-1," "3):}]=[{:(6,-2," "2),(-2," "3,-1),(2,-1," "3):}]`
`thereforeA+A=[{:(6,-2," "2),(-2," "3,-1),(2,-1," "3):}]+[{:(6,-2," "2),(-2," "3,-1),(2,-1," "3):}]`
`=[{:(12,-4," "4),(-4," "6,-2),(4,-2," "6):}]`
`rArr(1)/(2)(A+A')=[{:(6,-2," "2),(-2," "3,-1),(2,-1," "3):}]`
तथा `A-A'=[{:(6,-2," "2),(-2," "3,-1),(2,-1," "3):}]-[{:(6,-2," "2),(-2," "3,-1),(2,-1," "3):}]`
`=[{:(0,0,0),(0,0,0),(0,0,0):}]`
`rArr(1)/(2)(A-A')=[{:(0,0,0),(0,0,0),(0,0,0):}]`
अब `A=(1)/(2)(A+A')+(1)/(2)(A-A')`
`rArr[{:(6,-2," "2),(-2," "3,-1),(2,-1," "3):}]=[{:(6,-2," "2),(-2," "3,-1),(2,-1," "3):}]+[{:(0,0,0),(0,0,0),(0,0,0):}]`
(iii) माना `A=[{:(3,3,-1),(-2,-2," "1),(-4,-5," "2):}]`
`rArrA'=[{:(3,3,-1),(-2,-2," "1),(-4,-5," "2):}]=[{:(3,-2,-4),(3,-2,-5),(-1," "1," "2):}]`
`thereforeA+A'=[{:(3,3,-1),(-2,-2," "1),(-4,-5," "2):}]+[{:(3,-2,-4),(3,-2,-5),(-1," "1," "2):}]`
`=[{:(6,1,-5),(1,-4,-4),(-5,-4," "4):}]`
`rArr(1)/(2)(A+A')=[{:(3,(1)/(2),(-5)/(2)),((1)/(2),-2,-2),((-5)/(2),-2," "2):}]`
तथा `A-A'=[{:(3,3,-1),(-2,-2," "1),(-4,-5," "2):}]-[{:(3,-2,-4),(3,-2,-5),(-1," "1," "2):}]`
`=[{:(0,5,3),(-5,0,6),(-3,-6,0):}]`
`rArr(1)/(2)(A-A')=[{:(0,(5)/(2),(3)/(2)),(-(5)/(2),0,3),(-(3)/(2),-3,0):}]`
अब, `A=(1)/(2)(A+A')+(1)/(2)(A-A')`
`[{:(3,3,-1),(-2,-2," "1),(-4,-5," "2):}]=[{:(3,(1)/(2),(-5)/(2)),((1)/(2),-2,-2),((-5)/(2),-2,-2):}]+[{:(0,(5)/(2),(3)/(2)),((-5)/(2),0,3),((-3)/(2),-3,0):}]`
(iv) माना `A=[{:(1,5),(-1,2):}]`
`rArrA'=[{:(1,5),(-1,2):}]=[{:(1,-1),(5," " 2):}]`
`thereforeA+A'=[{:(1,5),(-1,2):}]+[{:(1,-1),(5," "2):}]=[{:(2,4),(4,4):}]`
`rArr(1)/(2)(A+A')=[{:(1,2),(2,2):}]`
तथा `A-A'=[{:(1,5),(-1,2):}]-[{:(1,-1),(5,2):}]=[{:(0,6),(-6,0):}]`
`rArr(1)/(2)(A-A')=[{:(0,3),(-3,0):}]`
अब `A=(1)/(2)(A+A')+(1)/(2)(A-A')`
`rArr[{:(1,5),(-1,2):}]=[{:(1,2),(2,2):}]+[{:(0,3),(-3,0):}]`