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Definition of determinant for 2x2 matrix

  1. Nov 3, 2014 #1
    For a 2x2 matrix, does the general definition hold?
    c131bc216fed2b3333b639a10cfb7d18.png

    If so, how exactly is the minor ## M_{i,j} ## computed in this case? If A is a 2x2 matrix, is det(A) only defined as ad - bc?
     
  2. jcsd
  3. Nov 3, 2014 #2

    Mark44

    Staff: Mentor

    The minor in this case is 1 x 1 matrix determinant. IOW, the number that isn't in row i, column j.

    $$\det \begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc$$
     
  4. Nov 4, 2014 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    No, that definition does not hold for any matrix because the determinant is a number where, as you have it, it is a function of "i".
    What you should have is a specific number in place of "i". Typically it is "1" but any number from 1 to n would do.

    In the case of a 2 by 2 matrix, n= 2 so, with your "i" replaced by "1", [itex]\sum_{j= 1}^N (-1)^{1+ j}a_{1j}M_{1j}[/itex] is [itex](-1)^{1+ 1}a_{11}a_{22}+ (-1)^{1+ 2}a_{12}a_{21}= a_{11}a_{22}- a_{12}a_{21}[/itex]. In the case that
    [tex]\begin{bmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{bmatrix}= \begin{bmatrix}a & b \\ c & d \end{bmatrix}[/tex]
    that would be ad- bc.

    Notice that if we replaced "1" above by "2" we would have [itex]\sum_{j= 1}^N (-1)^{2+ j}a_{2j}M_{2j}[/itex] is [itex](-1)^{2+ 1}a_{21}a_{12}+ (-1)^{2+ 2}a_{22}a_{11}= -a_{12}a_{21}+ a_{11}a_{22}[/itex] the same as before.

    The "minor", [itex]M_{ij}[/itex] is, by definition, the determinant of the matrix you have after removing the "ith row" and "jth column". In the case that
    [tex]A= \begin{bmatrix} a & b \\ c & d\end{bmatrix}[/tex]
    [itex]a_{11}= a[/itex] and [itex]M_{11}= a_{22}= d[/itex] since we remove the first row, [itex]\begin{bmatrix}a & b\end{bmatrix}[/itex] and first column, [itex]\begin{bmatrix}a \\ c \end{bmatrix}[/itex] we are left with the "1 by 1 matrix", d.
    Similarly, [itex]a_{12}= b[/itex] and [itex]M_{12}= a_{21}= c[/itex].
     
    Last edited: Nov 5, 2014
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