

Matrices
and Determinants 
The determinant of a
matrix 
Expanding a determinant
by cofactors

Calculating the value
of a determinant 
The
determinant of a 2by2
matrix 
The
determinant of a 3by3
matrix 





The
determinant of a matrix 
Determinants
are defined only for square matrices. 
The
determinant D
of an nbyn
matrix A,
also denoted det(A)
or A
is an ordered square array of elements 

the
value of which is given by an alternating sum of products of the
elements of A,
that can be obtained 
by
using the method of expanding the determinant to
cofactors. 

Expanding a determinant
by cofactors

That
is, the value of a determinant equals the sum of the products of
the entries in anyone row or column and their
respective cofactors. 
A cofactor
of an entry a_{ik}
is
a signed minor or (a subdeterminant) derived from a given matrix or
determinant
by the deletion of the row and column containing the specified
entry. 
Therefore, the
cofactor
of the entry a_{ik}
of a matrix A 
A_{ik} =
(1)^{i
+ k} · D_{ik
}, 
where
D_{ik}
is the minor (or subdeterminant) of the matrix A
obtained by deleting its i^{th}
row and k^{th}
column. 
A
determinant can be expanded along any row or column without
affecting the determinant's value. 
Thus,
the value of a determinant of order n
expanded along the k^{th}
column 

where
k
can be any column between 1 and n. 

Calculating the value
of a determinant 
The
determinant of a 2by2
matrix 
Let
apply the above formula to calculate the value of the
determinant of a 2
´
2
matrix A, 

then,
the expansion of the above determinant by the first column 

thus,
obtained is 


The
determinant of a 3by3
matrix 
Using
the above determinant expansion by cofactors formula we calculate the value of the
determinant of a 3
´
3
matrix A, 

The
diagram below shows the method of determining cofactors of the 2^{nd}
column's entries, as we chose to expand
the above determinant along that column 

thus,
obtained is 


Example:
For the given 3by3
matrix
A, find the
value of the determinant D
= det(A),


By
expanding the determinant by the first column obtained is 









Matrices, determinants and systems of linear equations
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