Determinant of a matrix
Determinants palys an important role in finding the inverse of a square matrix and also in solving systems of linear equations.
Determinant of a 2*2 matrix
Assuming A is an arbitrary 2*2 matrix A, where the elements are given by:
A = {{
![$a_{11}$](/wiki/pub/Main/QcDeterminant/latexeaabc9ca9a9cca9cd42537ad895adfb2.png)
,
![$a_{12}$](/wiki/pub/Main/QcDeterminant/latex51db1f0e70f66106fbf40836adedad0a.png)
},{
![$a_{21}$](/wiki/pub/Main/QcDeterminant/latex6705a0baa7decbb1a8ed5ae9032eee3d.png)
,
![$a_{22}$](/wiki/pub/Main/QcDeterminant/latexdfe42d8dd70e601213e1aec9c9ee7fae.png)
}}
then the determinant of a this matrix is as the follows:
det(A) = |A| = |{
![$a_{11}$](/wiki/pub/Main/QcDeterminant/latexeaabc9ca9a9cca9cd42537ad895adfb2.png)
,
![$a_{12}$](/wiki/pub/Main/QcDeterminant/latex51db1f0e70f66106fbf40836adedad0a.png)
},{
![$a_{21}$](/wiki/pub/Main/QcDeterminant/latex6705a0baa7decbb1a8ed5ae9032eee3d.png)
,
![$a_{22}$](/wiki/pub/Main/QcDeterminant/latexdfe42d8dd70e601213e1aec9c9ee7fae.png)
}| =
![$a_{11}$](/wiki/pub/Main/QcDeterminant/latexeaabc9ca9a9cca9cd42537ad895adfb2.png)
![$a_{12}$](/wiki/pub/Main/QcDeterminant/latex51db1f0e70f66106fbf40836adedad0a.png)
-
![$a_{21}$](/wiki/pub/Main/QcDeterminant/latex6705a0baa7decbb1a8ed5ae9032eee3d.png)
Determinant of a 3*3 matrix
The determinant of a 3*3 matrix is found as follows
A = {{
![$a_{11}$](/wiki/pub/Main/QcDeterminant/latexeaabc9ca9a9cca9cd42537ad895adfb2.png)
,
![$a_{12}$](/wiki/pub/Main/QcDeterminant/latex51db1f0e70f66106fbf40836adedad0a.png)
,
![$a_{13}$](/wiki/pub/Main/QcDeterminant/latex48d5c46fe48b8708f9c8eabefd2b2142.png)
},{
![$a_{21}$](/wiki/pub/Main/QcDeterminant/latex6705a0baa7decbb1a8ed5ae9032eee3d.png)
,
![$a_{22}$](/wiki/pub/Main/QcDeterminant/latexdfe42d8dd70e601213e1aec9c9ee7fae.png)
,
![$a_{23}$](/wiki/pub/Main/QcDeterminant/latex7076d07e5687568bff4d24f62ed0f2e6.png)
},{
![$a_{31}$](/wiki/pub/Main/QcDeterminant/latex25539e393a7357b9b7b93243e157bf16.png)
,
![$a_{32}$](/wiki/pub/Main/QcDeterminant/latex3282c349e602fcefa6a716a977019728.png)
,
![$a_{33}$](/wiki/pub/Main/QcDeterminant/latex62481be00a7b813b248fa83dabb89cd5.png)
}}
then the determinant of a this matrix is as follows:
det(A) = |A| = |{
![$a_{11}$](/wiki/pub/Main/QcDeterminant/latexeaabc9ca9a9cca9cd42537ad895adfb2.png)
,
![$a_{12}$](/wiki/pub/Main/QcDeterminant/latex51db1f0e70f66106fbf40836adedad0a.png)
,
![$a_{13}$](/wiki/pub/Main/QcDeterminant/latex48d5c46fe48b8708f9c8eabefd2b2142.png)
},{
![$a_{21}$](/wiki/pub/Main/QcDeterminant/latex6705a0baa7decbb1a8ed5ae9032eee3d.png)
,
![$a_{22}$](/wiki/pub/Main/QcDeterminant/latexdfe42d8dd70e601213e1aec9c9ee7fae.png)
,
![$a_{23}$](/wiki/pub/Main/QcDeterminant/latex7076d07e5687568bff4d24f62ed0f2e6.png)
},{
![$a_{31}$](/wiki/pub/Main/QcDeterminant/latex25539e393a7357b9b7b93243e157bf16.png)
,
![$a_{32}$](/wiki/pub/Main/QcDeterminant/latex3282c349e602fcefa6a716a977019728.png)
,
![$a_{33}$](/wiki/pub/Main/QcDeterminant/latex62481be00a7b813b248fa83dabb89cd5.png)
}|
=
![$a_{11}$](/wiki/pub/Main/QcDeterminant/latexeaabc9ca9a9cca9cd42537ad895adfb2.png)
|{
![$a_{22}$](/wiki/pub/Main/QcDeterminant/latexdfe42d8dd70e601213e1aec9c9ee7fae.png)
,
![$a_{23}$](/wiki/pub/Main/QcDeterminant/latex7076d07e5687568bff4d24f62ed0f2e6.png)
},{
![$a_{32}$](/wiki/pub/Main/QcDeterminant/latex3282c349e602fcefa6a716a977019728.png)
,
![$a_{33}$](/wiki/pub/Main/QcDeterminant/latex62481be00a7b813b248fa83dabb89cd5.png)
}|-
![$a_{12}$](/wiki/pub/Main/QcDeterminant/latex51db1f0e70f66106fbf40836adedad0a.png)
|{
![$a_{21}$](/wiki/pub/Main/QcDeterminant/latex6705a0baa7decbb1a8ed5ae9032eee3d.png)
,
![$a_{23}$](/wiki/pub/Main/QcDeterminant/latex7076d07e5687568bff4d24f62ed0f2e6.png)
},{
![$a_{31}$](/wiki/pub/Main/QcDeterminant/latex25539e393a7357b9b7b93243e157bf16.png)
,
![$a_{33}$](/wiki/pub/Main/QcDeterminant/latex62481be00a7b813b248fa83dabb89cd5.png)
}|+
![$a_{13}$](/wiki/pub/Main/QcDeterminant/latex48d5c46fe48b8708f9c8eabefd2b2142.png)
|{
![$a_{21}$](/wiki/pub/Main/QcDeterminant/latex6705a0baa7decbb1a8ed5ae9032eee3d.png)
,
![$a_{22}$](/wiki/pub/Main/QcDeterminant/latexdfe42d8dd70e601213e1aec9c9ee7fae.png)
},{
![$a_{31}$](/wiki/pub/Main/QcDeterminant/latex25539e393a7357b9b7b93243e157bf16.png)
,
![$a_{32}$](/wiki/pub/Main/QcDeterminant/latex3282c349e602fcefa6a716a977019728.png)
}|
Determinant of a n*n matrix
For the general case, where A is an n*n matrix the determinant is given by:
det(A) = |A| =
![$a_{11}\alpha_{11}$](/wiki/pub/Main/QcDeterminant/latex8d315acb2679235aa86b469fdd15529b.png)
+
![$a_{12}\alpha_{12}$](/wiki/pub/Main/QcDeterminant/latexd63c4810355b135f9b4f308dc384df6c.png)
+...+
![$a_{1n}\alpha_{1n}$](/wiki/pub/Main/QcDeterminant/latexc9622f70f044b05a5654d349f1b0c935.png)
Where the coefficients
![$\alpha_{ij}$](/wiki/pub/Main/QcDeterminant/latex6bd4087597e47d3115ddba025c8ff4fc.png)
are given by the relation
![$\alpha_{ij}$](/wiki/pub/Main/QcDeterminant/latex6bd4087597e47d3115ddba025c8ff4fc.png)
=
![$(-1)^{i+j}$](/wiki/pub/Main/QcDeterminant/latex9835ca1a34fc11466824272438df4cbe.png)
![$\beta_{ij}$](/wiki/pub/Main/QcDeterminant/latex8481f303a471fae826efae5c40a3c1ac.png)
where
![$\beta_{ij}$](/wiki/pub/Main/QcDeterminant/latex8481f303a471fae826efae5c40a3c1ac.png)
is the determinant of the (n-1)*(n-1) matrix that is obtained by deleting row i and column j. This coefficient
![$\alpha_{ij}$](/wiki/pub/Main/QcDeterminant/latex6bd4087597e47d3115ddba025c8ff4fc.png)
is also called the cofacotr of
Minor of a matrix
A minor of a matrix is the determiniant of a certain samller matrix. Suppose A is an m*n matrix and k is a positive integer not larger than m and n. A k*k minor of A is the determinant of a k*k matrix obtained from A by deleting m-k rows and n-k coulmns.
example:
given the matrix A = {{1,4,7}{3,0,5}{-1,9,11}}
the minor
![$M_{23}$](/wiki/pub/Main/QcDeterminant/latex56f71889353168add7864a03ccad61e7.png)
of A is
![$M_{23}$](/wiki/pub/Main/QcDeterminant/latex56f71889353168add7864a03ccad61e7.png)
= {{1,4}{-1,9}}
Adjoint of a matrix
Adjoint matrix can be calculated by the following method
- Given the n*n matrix A, define B = (
)
to be the matrix whose coefficients are found by taking the determinant of the (n-1)*(n-1) matrix obtained by deleting the ith row and jth column of A. The terms of B(i.e. B =
![$b_{ij}$](/wiki/pub/Main/QcDeterminant/latex36d2f83728875dad9a10609a39019dcc.png)
) are known as the cofactors of A.
- And define the matrix C, where
- The transpose of C is called the adjoint of Matrix A.