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Just starting: definition, and some useful properties
The '''determinant''' of a [[matrix]] (written |A|) is a single number that depends on the elements of the matrix A. Determinants only exist for square matrices (i.e., ones where the number of rows equals the number of columns).
==How to calculate==
We need to introduce two notions: the ''minor'' and the ''cofactor'' of a matrix element. Also, the determinant of a 1x1 matrix equals the sole element of that matrix.
;Minor: The minor <math>m_{ij}</math> of the element <math>a_{ij}</math> of an NxN matrix is the determinant of the (N-1)x(N-1) matrix formed by removing the ith row and jth column from M.
;Cofactor: The cofactor <math>C_{ij}</math> equals the minor <math>m_{ij}</math> multiplied by <math>(-1)^{i+j}</math>
The determinant is then defined to be the sum of the products of the elements of any one row or column with their corresponding cofactors.
===2x2 case===
For the 2x2 matrix
<math>\begin{pmatrix} a & b \\ c & d\end{pmatrix}</math>
the determinant is simply ad-bc (for example, using the above rule on the first row).
===3x3 case===
For a general 3x3 matrix
<math>\begin{pmatrix}
A_{11} & A_{12} & A_{13} \\
A_{21} & A_{22} & A_{23} \\
A_{31} & A_{32} & A_{33} \\
\end{pmatrix}</math>
we can expand along the first row to find
<math>
|A|=A_{11}\begin{vmatrix}A_{22} & A_{23} \\ A_{32} & A_{33} \end{vmatrix}-
A_{12}\begin{vmatrix}A_{21} & A_{23} \\ A_{31} & A_{33} \end{vmatrix}+
A_{13}\begin{vmatrix}A_{21} & A_{22} \\ A_{31} & A_{32} \end{vmatrix}
</math>
where each of the 2x2 determinants is given above.
==Properties of determinants==
The following are some useful properties of determinants. Some are useful computational aids for simplifying the algebra needed to calculate a determinant. The first property is that <math>|M|=|M^T|</math> where the superscript "T" denotes transposition. Thus, although the following rules refer to the rows of a matrix they apply equally well to the columns.
* The determinant is unchanged by adding a multiple of one row to any other row.
* If two rows are interchanged the sign of the determinant will change
* If a common factor <math>\alpha</math> is factored out from each element of a single row, the determinant is multiplied by that same factor.
* If all the elements of a single row are zero (or can be made to be zero using the above rules) then the determinant is zero.
* <math>|AB|=|A||B|</math>
In practice, one of the most efficient ways of finding the determinant of a large matrix is to add multiples of rows and/or columns until the matrix is in triangular form such that all the elements above or below the diagonal are zero, for example
<math>\begin{pmatrix}
A_{11} & A_{12} & A_{13} \\
0 & A_{22} & A_{23} \\
0 & 0 & A_{33} \\
\end{pmatrix}</math>.
The determinant of such a matrix is simply the product of the diagonal elements (use the cofactor expansion discussed above and expand down the first column).
[[category:mathematics]]
==How to calculate==
We need to introduce two notions: the ''minor'' and the ''cofactor'' of a matrix element. Also, the determinant of a 1x1 matrix equals the sole element of that matrix.
;Minor: The minor <math>m_{ij}</math> of the element <math>a_{ij}</math> of an NxN matrix is the determinant of the (N-1)x(N-1) matrix formed by removing the ith row and jth column from M.
;Cofactor: The cofactor <math>C_{ij}</math> equals the minor <math>m_{ij}</math> multiplied by <math>(-1)^{i+j}</math>
The determinant is then defined to be the sum of the products of the elements of any one row or column with their corresponding cofactors.
===2x2 case===
For the 2x2 matrix
<math>\begin{pmatrix} a & b \\ c & d\end{pmatrix}</math>
the determinant is simply ad-bc (for example, using the above rule on the first row).
===3x3 case===
For a general 3x3 matrix
<math>\begin{pmatrix}
A_{11} & A_{12} & A_{13} \\
A_{21} & A_{22} & A_{23} \\
A_{31} & A_{32} & A_{33} \\
\end{pmatrix}</math>
we can expand along the first row to find
<math>
|A|=A_{11}\begin{vmatrix}A_{22} & A_{23} \\ A_{32} & A_{33} \end{vmatrix}-
A_{12}\begin{vmatrix}A_{21} & A_{23} \\ A_{31} & A_{33} \end{vmatrix}+
A_{13}\begin{vmatrix}A_{21} & A_{22} \\ A_{31} & A_{32} \end{vmatrix}
</math>
where each of the 2x2 determinants is given above.
==Properties of determinants==
The following are some useful properties of determinants. Some are useful computational aids for simplifying the algebra needed to calculate a determinant. The first property is that <math>|M|=|M^T|</math> where the superscript "T" denotes transposition. Thus, although the following rules refer to the rows of a matrix they apply equally well to the columns.
* The determinant is unchanged by adding a multiple of one row to any other row.
* If two rows are interchanged the sign of the determinant will change
* If a common factor <math>\alpha</math> is factored out from each element of a single row, the determinant is multiplied by that same factor.
* If all the elements of a single row are zero (or can be made to be zero using the above rules) then the determinant is zero.
* <math>|AB|=|A||B|</math>
In practice, one of the most efficient ways of finding the determinant of a large matrix is to add multiples of rows and/or columns until the matrix is in triangular form such that all the elements above or below the diagonal are zero, for example
<math>\begin{pmatrix}
A_{11} & A_{12} & A_{13} \\
0 & A_{22} & A_{23} \\
0 & 0 & A_{33} \\
\end{pmatrix}</math>.
The determinant of such a matrix is simply the product of the diagonal elements (use the cofactor expansion discussed above and expand down the first column).
[[category:mathematics]]