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Let A=[{:(1,-2),(5," "4),(3," "0):}],B=[...

Let `A=[{:(1,-2),(5," "4),(3," "0):}],B=[{:(" "3," "1),(" "0," "2),(-3," "5):}]" and "C=[{:(" "4," "3),(-2," "2),(" "1," "6):}].`
Verify that `(A+B)+C=A+(B+C).`

Text Solution

Verified by Experts

Clearly, each one of the martices A, B, C is a `(3xx2)` matrix. So, `(A+B)+C` and `A+(B+C)` are both defined and each one is a `3xx2` matrix.
Now, `(A+B)=[{:(1,-2),(5," "4),(3," "0):}]+[{:(3,1),(0,2),(-3,5):}]`
`=[{:(1+3,-2+1),(5+0," "4+2),(3+(-3)," "0+5):}]+[{:(4,-1),(5," "6),(0," "5):}].`
`:." "(A+B)+C=[{:(4,-1),(5," "6),(0," "5):}]+[{:(" "4," "3),(-2," "2),(" "1," "6):}]`
`=[{:(4+4,-1+3),(5+(-2)," "6+2),(0+1," "5+6):}]=[{:(8,2),(3,8),(1,11):}].`
Also, `(B+C)=[{:(3,1),(0,2),(-3,5):}]+[{:(4,3),(-2,2),(1,6):}]`
`=[{:(3+4,1+3),(0+(-2),2+2),(-3+1,5+6):}]=[{:(7,4),(-2,4),(-2,11):}]`
`:." "A+(B+C)=[{:(1,-2),(5," "4),(3," "0):}]+[{:(7,4),(-2,4),(-2,11):}]`
`=[{:(1+7,-2+4),(5+(-2)," "4+4),(3+(-2)," "0+11):}]=[{:(8,2),(3,8),(1,11):}].`
Hence, `(A+B)+C=A+(B+C).`
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RS AGGARWAL-MATRICES-Exercise 5F
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  2. Construct a 2xx3 matrix whose elements are given by a(ij)=(1)/(2)(i-...

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  3. Construct a 2xx3 matrix whose elements are given by a(ij)=(1)/(2)|-3...

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  4. If [{:(x+2y,-y),(3x,4):}]=[{:(-4,3),(6,4):}], find the values of x and...

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  5. Find the values of x and y, if 2[{:(1,3),(0,x):}]+[{:(y,0),(1,2) :}]...

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  6. If x.[{:(2),(3):}]+y.[{:(-1),(1):}]=[{:(10),(5):}], find the values of...

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  8. If [{:(x,6),(-1,2w):}]+[{:(" "4,x+y),(z+w," "3):}]=3[{:(x,y),(z,w):}],...

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  9. If A=" diag "(3" "-2" "5)" and "B=" diag "(1" "3" "-4)," find "(A+B).

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  11. If A=[{:(1,-5),(-3," "2),(4,-2):}]" and "B=[{:(3," "1),(2,-1),(-2," "3...

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  12. If A=[{:(cosalpha,-sinalpha),(sinalpha,cosalpha):}], then find the lea...

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  15. If A=[{:(4,5),(1,8):}], show that (A+A') is symmetric.

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  16. If A=[{:(2,3),(4,5):}], show that (A-A') is skew-symmetric.

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  17. If A=[{:(2,-3),(4," "5):}]" and "B=[{:(-1,2),(0,3):}], find a matrix X...

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  18. If A=[{:(4,2),(1,3):}]" and "B=[{:(-2,1),(3,2):}], find a matrix X suc...

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  19. If (i) A=[cosalphasinalpha-sinalphacosalpha] , then verify that Aprime...

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