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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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RD SHARMA ENGLISH-ALGEBRA OF MATRICES-All Questions
  1. If the matrix A=[[0,a,-3],[2,0,-1],[b,1,0]] is skew-symmetric, find ...

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  2. Let A be a square matrix. Then prove that (i) A + A^T is a symmetric m...

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

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  4. If A and B are symmetric matrices, then show that A B is symmetric i...

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  5. Show that the matrix B^T\ A B is symmetric or skew-symmetric accord...

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  6. Let A and B be symmetric matrices of same order. Then AB-BA is a skew ...

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  7. Express the matrix A=|(3, 2 ,3),(4,5 ,3),( 2, 4,5)| as the sum of a ...

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  8. Show that all positive integral powers of a symmetric matrix are sy...

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  9. Show that positive odd integral powers of a skew-symmetric matrix are ...

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  10. A matrix which is both symmetric as well as skew-symmetric is a nul...

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  11. If A=[[2, 3], [4, 5]] , prove that A-A^T is a skew-symmetric matrix.

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  12. If A=[[3,-4], [1,-1]] , show that A-A^T is a skew symmetric matrix.

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  13. If the matrix A=[[5 ,2,x], [y, z,-3], [4,t,-7]] is a symmetric matrix,...

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  14. Let A=|(3, 2, 7),( 1, 4, 3),(-2, 5, 8)| . Find matrices X and Y such t...

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  15. Express the matrix A=[[4, 2,-1], [3 ,5, 7], [1,-2 ,1]] as the sum of a...

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  16. Define a symmetric matrix. Prove that for A=[2 4 5 6] , A+A^T is...

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  17. Express the matrix A=[[3,-4], [1,-1]] as the sum of a symmetric and a ...

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  18. Express the matrix A=[[3,-2,-4], [3,-2,-5],[-1, 1, 2]] as the sum of a...

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  19. If A is an mxxn matrix and B is nxxp matrix does A B exist? If yes, wr...

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  20. If A=[[2, 1, 4], [4, 1, 5]] and B=[[3,-1], [2, 2], [1,3]] . Write the ...

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