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Determinant of a Square Matrix,
Matrix Determinant Formula


Prior to learning about how to establish the inverse of a given matrix, something to learn about is the ‘determinant‘ of a square matrix.

The determinant is a real number that can be obtained from the elements of a square matrix, it can be a positive number or a negative number.


To find the determinant of a square matrix of size  2 x 2  and  3 x 3  in particular,
the determinant can be obtained performing calculations using a matrix determinant formula.

The common notation for the determinant is,     det( A )  or   | A |.


While the determinant of a square matrix is a value that turns out to be useful in a few branches of Mathematics, it is especially key in finding the inverse of a matrix





Determinant of a 2 x 2 Matrix

Finding the determinant of a square matrix that is  2 × 2  in size is quite straightforward.


For a matrix    \begin{bmatrix} a & b \\ c & d \end{bmatrix},    the determinant is obtained by    adbc.



Examples    


(1.1) 

B  =  \begin{bmatrix} 3 & 4 \\ {\text{-}}1 & 5 \end{bmatrix}       ,       det( B )  =  (3 × 5) − (-1 × 4)  =  19



(1.2) 

C  =  \begin{bmatrix} 0 & 5 \\ 2 & 7 \end{bmatrix}       ,       det( C )  =  (0 × 7) − (2 × 5)  =  –10






Determinant of a Square Matrix, 3 x 3


There are a few ways one can find the determinant of a  3 × 3  matrix.

We’ll focus on the method referred to as ‘expansion using minors and cofactors’ here.

Sometimes just referred to as expansion using minors.
Using this method with a (3 × 3) matrix involves breaking the matrix down into smaller (2 × 2) matrices and using them in a specific way.




Expansion using Minors and Cofactors Process



Minors:

If we have a standard  3 × 3  matrix:

A  =  \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}


We’ll initially focus on making use of the element  a_{11}.
If we erase the rest of the row and column elements in which  a_{11}  sits, we are left with a smaller (2 × 2) matrix.

First step to take in finding the determinant of a square matrix of size 3 x 3.


This new smaller (2 × 2) matrix is a ‘minor’ matrix, the notation for which is  M_{11}.

M_{11} = \begin{bmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{bmatrix}

We can also do the same with elements  a_{12}  and  a_{13}  to get the follwoing.

M_{12} = \begin{bmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{bmatrix}     M_{13} = \begin{bmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{bmatrix}.


Each of these (2 × 2) minor matrices will have it’s own determinant.     det(M_{11})     ,     det(M_{12})     ,     det(M_{13})




Cofactors:

Now the ‘cofactor’  C  of a matrix element  a_{ij}  is a number which is found by:

\boldsymbol{C_{ij}} =  \boldsymbol{({\text{-}}1)^{i+j}}det\boldsymbol{(M_{ij})}.

Thus the cofactor of  a_{11}  would be,   \boldsymbol{C_{11}} =  \boldsymbol{({\text{-}}1)^{1+1}}det\boldsymbol{(M_{11})}



Now we can look to put this all together.

If  A  is an  n × n  matrix. Then the determinant  det(A)  is given by:

    det(A)   =   {\displaystyle\sum_{j=1}^n} a_{ij}({\text{-}}1)^{i+j}det(M_{ij})


So for our (3 × 3) matrix A, using the 1st row a entries, so i = 1, we have:

det(A)   =   {\displaystyle\sum_{j=1}^3} a_{1j}({\text{-}}1)^{i+j}det(M_{1j})

=   {a_{11}({\text{-}}1)^{1+1}}det(M_{11})  +  {a_{12}({\text{-}}1)^{1+2}}det(M_{12})  +  {a_{13}({\text{-}}1)^{1+3}}det(M_{13})

=   {a_{11}}det(M_{11})  −  {a_{12}}det(M_{12})  +  {a_{13}}det(M_{13})



The result we’ve ended up with above is the formula we can use to find the determinant of a square matrix of size  3 × 3.

This can be a lot to take in at first, but let’s see this formula in action with an example of a matrix involving numbers.



Example    


(2.1) 

Find the determinant of the matrix

G  =  \begin{bmatrix} 3 & 7 & 2 \\ 2 & {\text{-}}1 & 6 \\ 0 & 4 & 5 \end{bmatrix}.

Solution   

We’ll again use the top row of elements to obtain our minor matrices.

a_{11} = 3   ,   a_{12} = 7   ,   a_{13} = 2


det(G)   =   {a_{11}}det(M_{11})  −  {a_{12}}det(M_{12})  +  {a_{13}}det(M_{13})


det(G)   =   (3 × det\begin{bmatrix} {\text{-}}1 & 6 \\ 4 & 5 \end{bmatrix})  −  (7 × det\begin{bmatrix} 2 & 6 \\ 0 & 5 \end{bmatrix})  +  (2 × det\begin{bmatrix} 2 & {\text{-}}1 \\ 0 & 4 \end{bmatrix})

=   (3 × (-29))  −  (7 × 10)  +  (2 × 8)

=   –8770 + 16  =  –141



You can actually go along any row or column of a matrix performing the same method with minor matrices and get the same determinant value for the matrix.

Why not try it with this and other 3 x 3 matrices and see.







Determinants of Larger Matrices


While the approach shown on this page will work with any size of square matrix, once you get to matrices of size 4 x 4 and larger, the calculations can become quite time consuming.

For larger matrices we usually leave the work of finding the determinant to a computer or a matrix determinant calculator.


A good matrix determinant calculator can be found here at the symbolab website.





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