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If P=[{:(x,0,0),(0,y,0),(0,0,z):}] and Q...

If `P=[{:(x,0,0),(0,y,0),(0,0,z):}]` and `Q=[{:(a,0,0),(0,b,0),(0,0 ,c):}]` then prove that
`PQ=[{:(xa,0,0),(0,yb,0),(0,0,zc):}]=QP`.

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`PQ=[{:( x,0,0),(0,y,0),(0,0,z):}][{:(a,0,0),(0,b,0),(0,0, c):}]=[{:(xa,0,0),(0,yb,0),(0,0,zc):}]`
and `QP=[{:(a,0,0),(0,b,0),(0,0,c):}] [{:(x,0,0),(0,y,0),(0,0,y):}]=[{:(ax ,0,0),(0,by,0),(0,0,zc):}]`
Thus, we see that `PQ=QP`
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NCERT EXEMPLAR-MATRICES-Matrices
  1. Give an example of three matrices A ,\ B ,\ C such that A B=A C but...

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  2. If A=[{:(1,2),(-2,1):}],B=[{:(2,3),(3,-4):}] and C=[{:(1,0),(-1,0):}],...

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  3. If P=[{:(x,0,0),(0,y,0),(0,0,z):}] and Q=[{:(a,0,0),(0,b,0),(0,0 ,c):}...

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  4. If [2 1 3] [{:(-1,0,-1),(-1,1,0),(0,1,1):}][{:(1),(0),(-1):}]=A, then ...

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  5. If A=[{:(2, 1,),(4,2,)],B =[{:(2,3,4),(1,4,0):}] and C=[{:(-1,2,1),(1,...

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  6. If A=[{:(1,0,-1),(2,1,3),(0,1, 1):}] then verify that A^(2)+A=(A+I) , ...

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  7. If A=[{:(0,-1,2),(4,3,-4):}] and B=[{:(4,0),(1,3),(2,6):}] then verify...

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  8. If A=[{:(1,2),(4,1),(5,6):}] and B=[{:(1,2),(6,4),(7,3):}], then varif...

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  9. Show that A' A and A A' are both symmetric matrices for any matrix A.

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  10. Let A and B be square matrices of the order 3xx3 . Is (A B)^2=A^2B^...

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  11. Show that , if A and B are square matrices such that AB=BA, then (A+B)...

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  12. If A=[{:(1,2),(-1,3):}]B=[{:(4,0),(1,5):}],C=[{:(2,0),(1,-2):}] a=4 an...

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  13. If A=[{:(cos q,sin q),(-sin q, cos q):}] , then variefy that A^(2)=[{:...

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  14. If A=[{:( 0,-x),(x,0):}].B=[{:(0,1),(1,0):}] and x^(2)=-1 , then show ...

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  15. Verify that A^(2)=I, when A=[{:(0,1,-1),(4,-3,4),(3,-3,4):}]

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  16. If A is a square matrix, using mathematical induction prove that (A...

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  17. Find inverse, by elementary row operations (if possible) , of both fo...

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  18. If [{:(xy,4),(z+6,x+y):}]=[{:(8,w),(0,6):}], then find the values of x...

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  19. If A=[{:(1,5),(7,12):}] and B=[{:(9,1),( 7,8):}] then find a matrix C ...

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  20. If A=[{:(3,-5),(-1,2):}] then find A^(2)-5A- 4I.

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