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Show that all positive integral powers o...

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

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Solution: Let `A` is a symmetric matrix, then, we have to prove,
`(A^{n})^{T}=A^{n}`
We know, `(A B)^{T}=B^{T} A^{T}`
`therefore(A^{n})^{T}=(A cdot A cdot ldots cdot A)^{T}=A^{T} cdot A^{T} cdot A^{T} ldots A^{T}=(A^{T})^{n}=A^{n} `
`Rightarrow(A^{n})^{T}=A^{n} .`
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RD SHARMA-ALGEBRA OF MATRICES-Solved Examples And Exercises
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  2. Express the matrix A=[[3,2, 3],[ 4, 5, 3],[ 2, 4, 5]] as the sum of a ...

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

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

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

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

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

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  10. Express the matrix A=[4 2-1 3 5 7 1-2 1] as the sum of a symmetric and...

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

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  12. Express the matrix A=[3-4 1-1] as the sum of a symmetric and a skew-sy...

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  13. Express the matrix [3-2-4 3-2-5-1 1 2] as the sum of a symmetric and s...

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

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  15. If A=[2 1 4 4 1 5] and B=[3-1 2 2 1 3] . Write the orders of A B and B...

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  16. If A=[4 3 1 2] and B=[-4 3] , write A B .

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  17. If A=[1 2 3] , write AA^T .

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  18. Give an example of two non-zero 2xx2 matrices A and B such that A B...

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  19. If A=[2 3 5 7] , find A+A^T .

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  20. If A=[i0 0i] , write A^2 .

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