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Show that A' A and A A' are both symmetr...

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

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To show that \( A^T A \) and \( A A^T \) are both symmetric matrices for any matrix \( A \), we will follow these steps: ### Step 1: Show that \( A^T A \) is symmetric 1. **Define the matrix**: Let \( P = A^T A \). 2. **Find the transpose of \( P \)**: \[ P^T = (A^T A)^T ...
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NCERT EXEMPLAR ENGLISH-MATRICES-Solved example
  1. If A=[{:(0,-1,2),(4,3,-4):}] and B=[{:(4,0),(1,3),(2,6):}] then verify...

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

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

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

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

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

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

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

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

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

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

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

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

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

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  15. Find the values of a,b,c and d, if 3[{:(a,b),(c,d):}]=[{:(a,6),(-1,2...

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  16. Find the matrix A such that [{:(2,-1), (1,0),(-3,4):}]A=[{:(-1,-8,-10)...

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  17. If A=[{:(1,2),(4,1):}] , then find A^(2)+2A+7I.

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  18. If A=[{:(cos alpha,sin alpha),(-sin alpha, cos alpha):}] and A^(-1)=A'...

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  19. If matrix [{:(0,a,3),(2,b,-1),(c,1,0):}] is skew-symmetric matrix, the...

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  20. If P(x)=[(cosx, sinx),(-sinx, cosx)], then show that P(x).P(y)=P(x+y)=...

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