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I could not understand the concept while googling. can anybody provide help?

what will be the determinant of the following matrix?

$$ \left[\begin{array}{cccc} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{array} \right]$$

Thanks.... :)

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10  
Well, the matrix is not even square. –  Qiaochu Yuan Feb 22 '11 at 11:34
3  
To clarify, the determinant is only defined for matrices which are square, that is, the number of rows and columns have to be the same. –  Calle Feb 22 '11 at 12:34
    
Under en.wikipedia.org/wiki/Determinant_of_a_matrix there is a heading Example that works out a 3x3 determinant. –  Ross Millikan Feb 22 '11 at 14:00
    
Use Wolfram Alpha (when you get a square matrix). –  Yuval Filmus Feb 22 '11 at 15:58
    
@selvin: Did you want to do something else with this matrix, because the $\bigtriangleup$ is not defined, and hence neither is the inverse. So I am not sure what you would like to find from this example. It must be an $n \times n$ matrix before its possible to do any of those things. –  night owl Apr 17 '11 at 12:17
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5 Answers

up vote 5 down vote accepted

By the way, if you've continued your determinant (in order to draw a real one) in the same vein:

$$ D = \begin{vmatrix} 1 & 2 & 3 & 4 \\\ 5 & 6 & 7 & 8 \\\ 9 & 10 & 11 & 12 \\\ 13 & 14 & 15 & 16 \end{vmatrix} \ , $$

the answer would have been easy:

$$ D = \begin{vmatrix} 1 & 2 & 3 & 4 \\\ 5 & 6 & 7 & 8 \\\ 4 & 4 & 4 & 4 \\\ 4 & 4 & 4 & 4 \end{vmatrix} = 0 \ . $$

:-)

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how the last two row elements become all 4? i still dont u.stand...can u give a 3x3 example please? –  selvin Feb 23 '11 at 5:44
    
$13-9 = 4$, $14-10=4, \dots$, $9-5=4,\dots$ –  a.r. Feb 23 '11 at 8:28
1  
I forgot: en.wikipedia.org/wiki/Determinant . See "Properties". Number 4. –  a.r. Feb 23 '11 at 8:32
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So to show how to calculate the determinant of a $n \times n$ matrix can be shown in a multiple of ways, but the method that we will use here is expansion by minors. We will first start of with a $2 \times 2$ matrix and work our way to a $3 \times 3$ matrix.

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$

$\underline{Example~1:}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ $2 \times 2$ Case

$$\left[\begin{array}{ccc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right]$$

Idea is that you want to multiply the diagonals and subtract them. So in other words:

$$a_{11}\cdot a_{22}-a_{12}\cdot a_{21}$$

That will be the method for any $2 \times 2$ matrix. Now we will look at an example of a $3 \times 3$ matrix.

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$

$\underline{Example~2:}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ $3 \times 3$ Case

$$\left[ \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} \right]$$

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$

First we must know the following about determinants, otherwise using this method will subdue wrong results. Below are the algebraic sign of the elements position within the matrix:

$$\left[ \begin{array}{ccc} + & - & + \\ - & + & - \\ + & - & + \end{array} \right]$$

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ $\underline {\text{Evaluation of the determinant expanding it by the minors of column 1:}}$

= $~a_{11} \left[\begin{array}{ccc} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right] -a_{21} \left[\begin{array}{ccc} a_{12} & a_{13} \\ a_{32} & a_{33} \end{array} \right] + a_{31} \left[\begin{array}{ccc} a_{12} & a_{13} \\ a_{22} & a_{23} \end{array} \right] $

$\Rightarrow ~ a_{11} \left(a_{22}\cdot a_{33}-a_{23}\cdot a_{32} \right) -a_{21} \left(a_{12}\cdot a_{33}-a_{13}\cdot a_{32} \right) + ~a_{31} \left(a_{12}\cdot a_{23}-a_{13}\cdot a_{22} \right) $

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$

From here, you would just simply by multiply what inside the parenthesis and then subtracting the two quantities and then you would multiply the result inside of the parenthesis by the term outside the parenthesis for each (namely, $a_{11},~a_{21},~a_{31}$).

If you need more assistance on how to evaluate the determinant, here is a reference below:

1) Computing Determinants

Okay, well I hope this cleared it up some on how to calculate determinants for square matrices. There are other methods that you can perform for matrices that are larger say $4 \times 4$ or even $5 \times 5$ matrices. You would want to perform row operations for those or any other clever techniques you prefer.

Good~Luck.

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You should put your matrix in upper triangluar form then compute the product of the diagonal entries. This is much faster than expansion by cofactors.

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mathsadist indeed. –  Did Dec 12 '11 at 14:55
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Determinants are defined only for squared matrix. If you are sure about your question then simply the determinant for your matrix does not exist.

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Sunil: I don't mean to nitpick, but I think it is more precise to say that it is not defined for matrices other than nxn matrices. –  Herb Feb 23 '11 at 10:07
    
@Herb: Agreed :) –  Sunil Feb 23 '11 at 15:12
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