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Prove that every square matrix can be un...

Prove that every square matrix can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix.

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To prove that every square matrix can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix, we will follow these steps: ### Step 1: Define the square matrix Let \( A \) be any square matrix of order \( n \). ### Step 2: Express \( A \) as a sum of two matrices We can express \( A \) as: \[ ...
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CENGAGE ENGLISH-MATRICES-ILLUSTRATION
  1. Let A+2B=[(1,2,0),(6,-3,3),(-5,3,1)] and 2A-B=[(2,-1,5),(2,-1,6),(0,1,...

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  2. If [(lambda^(2)-2lambda+1,lambda-2),(1-lambda^(2)+3lambda,1-lambda^(2)...

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  3. Prove that every square matrix can be uniquely expressed as the sum of...

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  4. Matrix A ha s m rows and n+ 5 columns; matrix B has m rows and 11 - n ...

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

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  6. Find the value of x and y that satisfy the equations [(3,-2),(3,0),(2,...

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  7. Find the values of x ,\ \ y\ ,\ z if the matrix A=[0 2y z x y-z x-y z]...

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  8. If A=[costhetasintheta-sinthetacostheta], then prove that A^n=[cosnthe...

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  9. If A=((p,q),(0,1)), then show that A^(8)=((p^(8),q((p^(8)-1)/(p-1))),(...

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  10. Let A=[(2,1),(0,3)] be a matrix. If A^(10)=[(a,b),(c,d)] then prove th...

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  11. Show that the solutions of the equation [(x,y),(z,t)]^2=0 are[(x,y),(...

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  12. Let a be square matrix. Then prove that A A^(T) and A^(T) A are symmet...

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  13. If A, B are square materices of same order and B is a skewsymmetric ma...

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  14. If a and B are square matrices of same order such that AB+BA=O, then p...

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  15. Let A=[(1,2),(-1,3)] .If A^6=kA-205I then then numerical quantity of...

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  16. Let A, B, C, D be (not necessarily square) real matrices such that A^T...

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  17. If A and B are square matrices of the same order such that A B = B A,...

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  18. If A=[-1 1 0-2] , then prove that A^2+3A+2I=Odot Hence, find Ba n dC m...

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  19. If A=[(3,-4),(1,-1)] then find tr. (A^(2012)).

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  20. If A is a nonsingular matrix satisfying AB-BA=A, then prove that det. ...

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