Express the following matrices as the sum of a symmetric and a skew symmetricmatrix:(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]`

Answer

N/a

Explanation

`(i) Let A=[{:(3,5),(1,-1):}]`
`implies A'=[{:(3,5),(1,-1):}]=[{:(3,1),(5,-1):}]`
`therefore A+A'=[{:(3,5),(1,-1):}]+[{:(3,1),(5,-1):}]=[{:(6,6),(6,-2):}]`
`implies (1)/(2) (A+A')=[{:(3,3),(3,-1):}]`
`and A-A'=[{:(3,5),(1,-1):}]-[{:(3,1),(5,-1):}]=[{:(0,4),(-4,0):}]`
`implies (1)/(2) (A-A'=[{:(0,2),(-2,0):}]`
`Now, A=(1)/(2)(A+A')+(1)/(2)(A-A')`
` implies [{:(3,5),(1,-1):}]=[{:(3,3),(3,-1):}]+[{:(0,2),(-2,0):}]`
`(ii) Let A=[{:(6,-2,2),(-2,3,-1),(2,-1,3):}]`
`implies A'=[{:(6,-2,2),(-2,3,-1),(2,-1,
3):}]=[{:(6,-2,2),(-2,3,-1),(2,-1,3):}]`
`therefore A+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):}]`
`implies (1)/(2) (A+A')=[{:(6,-2,2),(-2,3,-1),(2,-1,3):}]`
`and A-A'=[{:(6,-2,2),(-2,3,-1),(2,-1,3):}]-[{:(6,-2,2),(2,-1,3):}]`
`=[{:(0,0,0),(0,0,0),(0,0,0):}]`
`implies(1)/(2)(A-A')=[{:(0,0,0),(0,0,0),(0,0,0):}]`
Now , `A=(1)/(2) (A+A')+(1)/(2) (a-A')`
`implies[{:(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) A=[{:(3,3,-1),(-2,-2,1),(-4,-5,2):}]`
`implies A'=[{:(3,3,-1),(-2,-2,1),(-4,-5,2):}]=[{:(3,-2,-4),(3,-2,-5),(-1,1,2):}]`
`thereofore A+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):}]`
`implies (1)/(2) (A+A')=[{:(3,(1)/(2),(-5)/(2)),((1)/(2),-2,-2),((-5)/(2),-2,-2):}]`
`and 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):}]`
`implies (1)/(2) (A-A')=[{:(0,(5)/(2),(3)/(2)),(-(5)/(2),0,3),(-(3)/(2),-3,0):}]`
now , `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) Let A=[{:(1,5),(-1,2):}]`
`implies A'=[{:(1,5),(-1,2):}]=[{:(1,-1),(5,2):}]`
`therefore A+A'=[{:(1,5),(-1,2):}]+[{:(1,-1),(5,2):}]=[{:(2,4),(4,4):}]`
`implies (1)/(2)(A+A')=[{:(1,2),(2,2):}]`
` and ,A-A=[{:(1,5),(-1,2):}]-[{:(1,-1):}]=[{:(0,6),(-6,0):}]`
`implies (1)/(2) (A-A')=[{:(0,3),(-3,0):}]`
`Now, A=(1)/(2)(A+A')+(1)/(2)(A-A')`
`implies[{:(1,5),(-1,2):}]+[{:(0,3),(-3,0):}]`