Show that $8$ is an eigen value of $A$

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Consider the following matrix:



$$A=beginbmatrixB&&&C\D&&&Fendbmatrix$$ where



$$B= beginbmatrix 9 &1&1&1&1\1&9&1&1&1\1&1&9&1&1\1&1&1&9&1\1&1&1&1&9endbmatrix$$ and $$C=beginbmatrix1&1&1&1&1\1&1&1&1&1\1&1&1&1&1\1&1&1&1&1\1&1&1&1&1endbmatrix$$



and $$D=C^T$$



and $$F=beginbmatrix G&H \I &Jendbmatrix$$



where $$G=beginbmatrix 8 &1&1&1\1&8&1&1\1&1&8&1\1&1&1&8endbmatrix$$ and $$H=beginbmatrix 0\0\0\0endbmatrix$$ and $$I=H^T$$ and



$$J=beginbmatrix 5endbmatrix$$




Show that $8$ is an eigen value of the matrix $A$.




How should I try to prove it?Please give some hints .I dont want a complete solution




linear-algebra matrices eigenvalues-eigenvectors




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    up vote
    4
    down vote

    favorite












    Consider the following matrix:



    $$A=beginbmatrixB&&&C\D&&&Fendbmatrix$$ where



    $$B= beginbmatrix 9 &1&1&1&1\1&9&1&1&1\1&1&9&1&1\1&1&1&9&1\1&1&1&1&9endbmatrix$$ and $$C=beginbmatrix1&1&1&1&1\1&1&1&1&1\1&1&1&1&1\1&1&1&1&1\1&1&1&1&1endbmatrix$$



    and $$D=C^T$$



    and $$F=beginbmatrix G&H \I &Jendbmatrix$$



    where $$G=beginbmatrix 8 &1&1&1\1&8&1&1\1&1&8&1\1&1&1&8endbmatrix$$ and $$H=beginbmatrix 0\0\0\0endbmatrix$$ and $$I=H^T$$ and



    $$J=beginbmatrix 5endbmatrix$$




    Show that $8$ is an eigen value of the matrix $A$.




    How should I try to prove it?Please give some hints .I dont want a complete solution




    linear-algebra matrices eigenvalues-eigenvectors




    share|cite|improve this question






















      up vote
      4
      down vote

      favorite









      up vote
      4
      down vote

      favorite











      Consider the following matrix:



      $$A=beginbmatrixB&&&C\D&&&Fendbmatrix$$ where



      $$B= beginbmatrix 9 &1&1&1&1\1&9&1&1&1\1&1&9&1&1\1&1&1&9&1\1&1&1&1&9endbmatrix$$ and $$C=beginbmatrix1&1&1&1&1\1&1&1&1&1\1&1&1&1&1\1&1&1&1&1\1&1&1&1&1endbmatrix$$



      and $$D=C^T$$



      and $$F=beginbmatrix G&H \I &Jendbmatrix$$



      where $$G=beginbmatrix 8 &1&1&1\1&8&1&1\1&1&8&1\1&1&1&8endbmatrix$$ and $$H=beginbmatrix 0\0\0\0endbmatrix$$ and $$I=H^T$$ and



      $$J=beginbmatrix 5endbmatrix$$




      Show that $8$ is an eigen value of the matrix $A$.




      How should I try to prove it?Please give some hints .I dont want a complete solution




      linear-algebra matrices eigenvalues-eigenvectors




      share|cite|improve this question












      Consider the following matrix:



      $$A=beginbmatrixB&&&C\D&&&Fendbmatrix$$ where



      $$B= beginbmatrix 9 &1&1&1&1\1&9&1&1&1\1&1&9&1&1\1&1&1&9&1\1&1&1&1&9endbmatrix$$ and $$C=beginbmatrix1&1&1&1&1\1&1&1&1&1\1&1&1&1&1\1&1&1&1&1\1&1&1&1&1endbmatrix$$



      and $$D=C^T$$



      and $$F=beginbmatrix G&H \I &Jendbmatrix$$



      where $$G=beginbmatrix 8 &1&1&1\1&8&1&1\1&1&8&1\1&1&1&8endbmatrix$$ and $$H=beginbmatrix 0\0\0\0endbmatrix$$ and $$I=H^T$$ and



      $$J=beginbmatrix 5endbmatrix$$




      Show that $8$ is an eigen value of the matrix $A$.




      How should I try to prove it?Please give some hints .I dont want a complete solution





      linear-algebra matrices eigenvalues-eigenvectors





      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Aug 23 at 3:39









      PureMathematics

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          Hint:



          • Examine the first two rows of $A-8I$, they are identical, what can you conclude.

          Same result holds if you replace $D$ and $F$ with other matrices of the same size.






          share|cite|improve this answer




















          • How to find the multiplicity,can you help
            – PureMathematics
            Aug 23 at 5:12











          • compute the nullity of $A-8I$, that is the multiplicity.
            – Siong Thye Goh
            Aug 23 at 5:18










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          oldest

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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          5
          down vote













          Hint:



          • Examine the first two rows of $A-8I$, they are identical, what can you conclude.

          Same result holds if you replace $D$ and $F$ with other matrices of the same size.






          share|cite|improve this answer




















          • How to find the multiplicity,can you help
            – PureMathematics
            Aug 23 at 5:12











          • compute the nullity of $A-8I$, that is the multiplicity.
            – Siong Thye Goh
            Aug 23 at 5:18














          up vote
          5
          down vote













          Hint:



          • Examine the first two rows of $A-8I$, they are identical, what can you conclude.

          Same result holds if you replace $D$ and $F$ with other matrices of the same size.






          share|cite|improve this answer




















          • How to find the multiplicity,can you help
            – PureMathematics
            Aug 23 at 5:12











          • compute the nullity of $A-8I$, that is the multiplicity.
            – Siong Thye Goh
            Aug 23 at 5:18












          up vote
          5
          down vote










          up vote
          5
          down vote









          Hint:



          • Examine the first two rows of $A-8I$, they are identical, what can you conclude.

          Same result holds if you replace $D$ and $F$ with other matrices of the same size.






          share|cite|improve this answer












          Hint:



          • Examine the first two rows of $A-8I$, they are identical, what can you conclude.

          Same result holds if you replace $D$ and $F$ with other matrices of the same size.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Aug 23 at 3:53









          Siong Thye Goh

          80.5k1453101




          80.5k1453101











          • How to find the multiplicity,can you help
            – PureMathematics
            Aug 23 at 5:12











          • compute the nullity of $A-8I$, that is the multiplicity.
            – Siong Thye Goh
            Aug 23 at 5:18
















          • How to find the multiplicity,can you help
            – PureMathematics
            Aug 23 at 5:12











          • compute the nullity of $A-8I$, that is the multiplicity.
            – Siong Thye Goh
            Aug 23 at 5:18















          How to find the multiplicity,can you help
          – PureMathematics
          Aug 23 at 5:12





          How to find the multiplicity,can you help
          – PureMathematics
          Aug 23 at 5:12













          compute the nullity of $A-8I$, that is the multiplicity.
          – Siong Thye Goh
          Aug 23 at 5:18




          compute the nullity of $A-8I$, that is the multiplicity.
          – Siong Thye Goh
          Aug 23 at 5:18

















           

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