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

*ad*−

*bc*.

__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.

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.

### 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**)

= –

**87**−

**70**+

**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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