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Math Help - Matricies Proof

  1. #1
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    Matricies Proof

    a) SHOW that a matrix with a row of zeros cannot have an inverse

    b) SHOW that a matrix with a column of zeros cannot have an inverse


    how do i show this? i have no clue
    help would be much appreciated
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  2. #2
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    Quote Originally Posted by Kbotz View Post
    a) SHOW that a matrix with a row of zeros cannot have an inverse

    b) SHOW that a matrix with a column of zeros cannot have an inverse


    how do i show this? i have no clue
    help would be much appreciated
    A matrix is invertible iff. the det\neq0. What happens if a column of row is alls zeros?

    Definition:
    The determinant of an nxn matrix A, denoted det(A), is a scalar associated with the matrix A that is defined inductively as
    det(A)=<br />
\begin{cases} <br />
  a_{11},  & \mbox{if }n=1 \\<br />
  a_{11}A_{11}+a_{12}A_{12}+\dots+a_{1n}A_{1n}, & \mbox{if }n>1 <br />
\end{cases}<br />
where A_{1j}=(-1)^{1+j}det(M_{1j}),\ j=1,...,n are the cofactors associated with the entries in the first row of A.

    Leon, S. (2010). Linear algebra with applications. Upper Saddle River, NJ: Pearson.

    a.
    By expanding across the row of all zeros, each term of the cofactor expansion will have a factor of 0. Hence, the sum will equal 0=det(A)
    Last edited by dwsmith; May 21st 2010 at 05:43 PM.
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  3. #3
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    so if a matrix consists of a column of zeros, it's det=0. do i prove it by solving for the determinant?
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  4. #4
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    Quote Originally Posted by Kbotz View Post
    so if a matrix consists of a column of zeros, it's det=0. do i prove it by solving for the determinant?
    That is what I would do, because when you solve the det of a matrix, you expand down the easiest row (usually the row with the most zeros).
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  5. #5
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    if i was to do it with any 3x3 matrix, i use the expansion of minors method.
    i can seem to do it for a column of zeros. how is it done for a row?
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  6. #6
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    Quote Originally Posted by Kbotz View Post
    if i was to do it with any 3x3 matrix, i use the expansion of minors method.
    i can seem to do it for a column of zeros. how is it done for a row?
     det(A)= <br />
\begin{cases} <br />
 a_{11}, & \mbox{if }n=1 \\ <br />
 a_{11}A_{11}+a_{12}A_{12}+\dots+a_{1n}A_{1n}, & \mbox{if }n>1 <br />
\end{cases}

     det(A)= <br />
\begin{cases} <br />
 a_{11}, & \mbox{if }n=1 \\ <br />
 a_{11}A_{11}+a_{21}A_{21}+\dots+a_{n1}A_{n1}, & \mbox{if }n>1 <br />
\end{cases}

    If we are expanding along the rows (first def) or the columns (second def), what are the values of a_{ij}?

    Or how about this if you can do it for columns.

    det(A)=det(A^T)
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  7. #7
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    thank you!
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