Results 1 to 7 of 7

Thread: Matricies Proof

  1. #1
    Junior Member
    Joined
    Apr 2010
    Posts
    30

    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
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    10
    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 $\displaystyle det\neq0$. 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
    $\displaystyle det(A)=
    \begin{cases}
    a_{11}, & \mbox{if }n=1 \\
    a_{11}A_{11}+a_{12}A_{12}+\dots+a_{1n}A_{1n}, & \mbox{if }n>1
    \end{cases}
    $ where $\displaystyle A_{1j}=(-1)^{1+j}det(M_{1j}),\ j=1,...,n$ 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 $\displaystyle 0=det(A)$
    Last edited by dwsmith; May 21st 2010 at 05:43 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Apr 2010
    Posts
    30
    so if a matrix consists of a column of zeros, it's $\displaystyle det=0$. do i prove it by solving for the determinant?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    10
    Quote Originally Posted by Kbotz View Post
    so if a matrix consists of a column of zeros, it's $\displaystyle det=0$. 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).
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Apr 2010
    Posts
    30
    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?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    10
    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?
    $\displaystyle det(A)=
    \begin{cases}
    a_{11}, & \mbox{if }n=1 \\
    a_{11}A_{11}+a_{12}A_{12}+\dots+a_{1n}A_{1n}, & \mbox{if }n>1
    \end{cases} $

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

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

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

    $\displaystyle det(A)=det(A^T)$
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Apr 2010
    Posts
    30
    thank you!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Matricies, need help...
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: Nov 26th 2010, 12:38 PM
  2. matricies
    Posted in the Algebra Forum
    Replies: 0
    Last Post: May 27th 2010, 05:57 PM
  3. 3x3 matricies proof
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Nov 20th 2009, 11:30 AM
  4. Matricies / equilibrium proof
    Posted in the Statistics Forum
    Replies: 1
    Last Post: May 27th 2009, 06:24 AM
  5. Matricies
    Posted in the Algebra Forum
    Replies: 1
    Last Post: Dec 18th 2008, 08:46 AM

Search Tags


/mathhelpforum @mathhelpforum