Find the determinant of the matrix.

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If
$textdetleft[beginmatrixa & 1 & c\
b & 1 & d\
e& 1 & f
endmatrixright]= -3$
and $textdetleft[beginmatrixa & 1 & c\
b & 2 & d\
e& 3 & f
endmatrixright]= 5$
find
$textdetleft[beginmatrixa & -4 & c\
b & -7 & d\
e& -10 & f
endmatrixright]$.



How do I approach this? The section deals with the effect of row operations on the determinate.






linear-algebra determinant







share|cite|improve this question

























    up vote
    2
    down vote

    favorite
    3












    If
    $textdetleft[beginmatrixa & 1 & c\
    b & 1 & d\
    e& 1 & f
    endmatrixright]= -3$
    and $textdetleft[beginmatrixa & 1 & c\
    b & 2 & d\
    e& 3 & f
    endmatrixright]= 5$
    find
    $textdetleft[beginmatrixa & -4 & c\
    b & -7 & d\
    e& -10 & f
    endmatrixright]$    .



    How do I approach this? The section deals with the effect of row operations on the determinate.






    linear-algebra determinant







    share|cite|improve this question























      up vote
      2
      down vote

      favorite
      3









      up vote
      2
      down vote

      favorite
      3






      3





      If
      $textdetleft[beginmatrixa & 1 & c\
      b & 1 & d\
      e& 1 & f
      endmatrixright]= -3$
      and $textdetleft[beginmatrixa & 1 & c\
      b & 2 & d\
      e& 3 & f
      endmatrixright]= 5$
      find
      $textdetleft[beginmatrixa & -4 & c\
      b & -7 & d\
      e& -10 & f
      endmatrixright]$      .



      How do I approach this? The section deals with the effect of row operations on the determinate.






      linear-algebra determinant







      share|cite|improve this question













      If
      $textdetleft[beginmatrixa & 1 & c\
      b & 1 & d\
      e& 1 & f
      endmatrixright]= -3$
      and $textdetleft[beginmatrixa & 1 & c\
      b & 2 & d\
      e& 3 & f
      endmatrixright]= 5$
      find
      $textdetleft[beginmatrixa & -4 & c\
      b & -7 & d\
      e& -10 & f
      endmatrixright]$      .



      How do I approach this? The section deals with the effect of row operations on the determinate.






      linear-algebra determinant




      linear-algebra determinant






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 3 hours ago









      johnny133253

      1479




      1479




















          1 Answer
          1






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          oldest

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          up vote
          3
          down vote



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          The determinant is linear in every row and in every column. Thus, this problem is equivalent to finding $φ(-4,-7,-10)$ for a linear map $φ colon ℚ^3 → ℚ$ with $φ(1,1,1) = -3$ and $φ(1,2,3)= 5$. Do you see that – what is $φ$ here? Can you solve this reduced problem?






          share|cite|improve this answer




















          • Makes perfect sense. Thanks. I just need to solve a system of equations.
            – johnny133253
            3 hours ago







          • 1




            @johnny133253 Yeah, or you just guess the linear combination $(-4,-7,-10) = μ(1,1,1) + λ(1,2,3)$.
            – k.stm
            3 hours ago










          • Yup, so $mu = -1$ and $lambda = -3$. Much appreciated.
            – johnny133253
            3 hours ago










          • @johnny133253,so have you got the value of determinant asked?
            – Dhamnekar Winod
            1 hour ago










          • Yes its $(-1)(-3) + (-3)(5) = -12$
            – johnny133253
            1 hour ago










          Your Answer




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          1 Answer
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          active

          oldest

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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          3
          down vote



          accepted










          The determinant is linear in every row and in every column. Thus, this problem is equivalent to finding $φ(-4,-7,-10)$ for a linear map $φ colon ℚ^3 → ℚ$ with $φ(1,1,1) = -3$ and $φ(1,2,3)= 5$. Do you see that – what is $φ$ here? Can you solve this reduced problem?






          share|cite|improve this answer




















          • Makes perfect sense. Thanks. I just need to solve a system of equations.
            – johnny133253
            3 hours ago







          • 1




            @johnny133253 Yeah, or you just guess the linear combination $(-4,-7,-10) = μ(1,1,1) + λ(1,2,3)$.
            – k.stm
            3 hours ago










          • Yup, so $mu = -1$ and $lambda = -3$. Much appreciated.
            – johnny133253
            3 hours ago










          • @johnny133253,so have you got the value of determinant asked?
            – Dhamnekar Winod
            1 hour ago










          • Yes its $(-1)(-3) + (-3)(5) = -12$
            – johnny133253
            1 hour ago














          up vote
          3
          down vote



          accepted










          The determinant is linear in every row and in every column. Thus, this problem is equivalent to finding $φ(-4,-7,-10)$ for a linear map $φ colon ℚ^3 → ℚ$ with $φ(1,1,1) = -3$ and $φ(1,2,3)= 5$. Do you see that – what is $φ$ here? Can you solve this reduced problem?






          share|cite|improve this answer




















          • Makes perfect sense. Thanks. I just need to solve a system of equations.
            – johnny133253
            3 hours ago







          • 1




            @johnny133253 Yeah, or you just guess the linear combination $(-4,-7,-10) = μ(1,1,1) + λ(1,2,3)$.
            – k.stm
            3 hours ago










          • Yup, so $mu = -1$ and $lambda = -3$. Much appreciated.
            – johnny133253
            3 hours ago










          • @johnny133253,so have you got the value of determinant asked?
            – Dhamnekar Winod
            1 hour ago










          • Yes its $(-1)(-3) + (-3)(5) = -12$
            – johnny133253
            1 hour ago












          up vote
          3
          down vote



          accepted







          up vote
          3
          down vote



          accepted






          The determinant is linear in every row and in every column. Thus, this problem is equivalent to finding $φ(-4,-7,-10)$ for a linear map $φ colon ℚ^3 → ℚ$ with $φ(1,1,1) = -3$ and $φ(1,2,3)= 5$. Do you see that – what is $φ$ here? Can you solve this reduced problem?






          share|cite|improve this answer












          The determinant is linear in every row and in every column. Thus, this problem is equivalent to finding $φ(-4,-7,-10)$ for a linear map $φ colon ℚ^3 → ℚ$ with $φ(1,1,1) = -3$ and $φ(1,2,3)= 5$. Do you see that – what is $φ$ here? Can you solve this reduced problem?







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 3 hours ago









          k.stm

          10.6k22249




          10.6k22249











          • Makes perfect sense. Thanks. I just need to solve a system of equations.
            – johnny133253
            3 hours ago







          • 1




            @johnny133253 Yeah, or you just guess the linear combination $(-4,-7,-10) = μ(1,1,1) + λ(1,2,3)$.
            – k.stm
            3 hours ago










          • Yup, so $mu = -1$ and $lambda = -3$. Much appreciated.
            – johnny133253
            3 hours ago










          • @johnny133253,so have you got the value of determinant asked?
            – Dhamnekar Winod
            1 hour ago










          • Yes its $(-1)(-3) + (-3)(5) = -12$
            – johnny133253
            1 hour ago
















          • Makes perfect sense. Thanks. I just need to solve a system of equations.
            – johnny133253
            3 hours ago







          • 1




            @johnny133253 Yeah, or you just guess the linear combination $(-4,-7,-10) = μ(1,1,1) + λ(1,2,3)$.
            – k.stm
            3 hours ago










          • Yup, so $mu = -1$ and $lambda = -3$. Much appreciated.
            – johnny133253
            3 hours ago










          • @johnny133253,so have you got the value of determinant asked?
            – Dhamnekar Winod
            1 hour ago










          • Yes its $(-1)(-3) + (-3)(5) = -12$
            – johnny133253
            1 hour ago















          Makes perfect sense. Thanks. I just need to solve a system of equations.
          – johnny133253
          3 hours ago





          Makes perfect sense. Thanks. I just need to solve a system of equations.
          – johnny133253
          3 hours ago





          1




          1




          @johnny133253 Yeah, or you just guess the linear combination $(-4,-7,-10) = μ(1,1,1) + λ(1,2,3)$.
          – k.stm
          3 hours ago




          @johnny133253 Yeah, or you just guess the linear combination $(-4,-7,-10) = μ(1,1,1) + λ(1,2,3)$.
          – k.stm
          3 hours ago












          Yup, so $mu = -1$ and $lambda = -3$. Much appreciated.
          – johnny133253
          3 hours ago




          Yup, so $mu = -1$ and $lambda = -3$. Much appreciated.
          – johnny133253
          3 hours ago












          @johnny133253,so have you got the value of determinant asked?
          – Dhamnekar Winod
          1 hour ago




          @johnny133253,so have you got the value of determinant asked?
          – Dhamnekar Winod
          1 hour ago












          Yes its $(-1)(-3) + (-3)(5) = -12$
          – johnny133253
          1 hour ago




          Yes its $(-1)(-3) + (-3)(5) = -12$
          – johnny133253
          1 hour ago

















           

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