Properties of Determinants

A determinant is a numerical value that can be obtained by solving a square matrix.  For a square matrix A, its determinant is denoted as det(A) or A. A square matrix is non-invertible if and only if its determinant is zero. Invertible matrices always have non-zero determinants 
 

The properties of determinants simplify the process of their computation. Take a look at some of these properties below:

Sum Property

When two columns or rows of a determinant are added, the determinant of a sum cannot be equal to the sum of determinants.  
For example: 
Let,
    

Then,
     

Repetition Property

If any two rows or columns of a determinant are identical or are multiples of one another, then the value of the determinant is zero.
For example:
           
Here, rows 1 and 3 are identical. So, det(A)=0

All zero property

For any row or column in a matrix, if all elements are zero, then the determinant is zero.
For example: 
        
             
Here, row 2 has all 0 elements. So A=0

Factor Property

According to the factor property of determinants, if substituting x as a in the determinant of a square matrix A results in zero, then (x - a) is a factor of the determinant expression.
For example, consider the determinant of the matrix:
          
Det = xx-32=x2-6
Set the determinant to zero, 
x2-6=0
 x= 6
Apply the factor theorem
Since x = 6 is a root, then (x-6) is a factor similarly  (x+6) is also a factor 
So, x2-6=(x-6)  (x+6)
This confirms that if a value of x makes the determinant zero, then (x - a) is a factor of the determinant.

Switching Property

If any two rows or columns of a determinant are interchanged, the sign of the determinant will also change, but the magnitude remains the same.
Mathematically, if A is a square matrix and A’ is the new matrix formed after interchanging 2 rows or columns, then det(A’) = -det(A)
For example, let 
        
det(A) = (2)(4) - (1)(3) = 8 - 3 = 5
Now, if we interchange the rows, 
       
det(A’) = (3)(1) - (4)(2) = 3 - 8 = -5
Proving that det(A’) = -det(A) when 2 rows or columns are interchanged.

Scalar Multiple Property

For any row/column multiplied by a non-zero scalar value k, the determinant will also be multiplied by k.
For example:
Let,
     A = 1  2 = (1)(4) - (2)(3) = 4-6 = -2
               3  4
Multiply row 1 by 5:
   A' = 5  10 = (5)(4)-(10)(3) = 20- 30 = -10
              3    4
We see that, A' =5 A = 5(-2) = -10

Triangle property

If elements above or below the main diagonal are zero, the determinant is the product of diagonal elements. 
For example: 
    For an upper triangular matrix, 
                                   
All elements below the diagonal are zero.
So, A = 2 × 3 × 6 = 36

Transpose of determinant/Reflection property

The transpose of a matrix is denoted by |AT| for any determinant |A|. This property suggests that the determinant remains unchanged on its transpose, i.e., |AT| =  |A|.
For example:
  Let:
          
Find the transpose AT:
           
Now find both determinants:
A = 1(59-68)- 2(49-67)+3(48-57) = 0
|AT| = 1(59-68)-4(29-38) + 7(26-35)=0
|AT| = |A| = 0
|AT| = |A| is valid even for non-zero values of determinants.

Determinant of Cofactor Matrix

For a square matrix A of order 2,
                 
The determinant is,
    A = 4  7 = (4)(6)-(7)(2)=24-14=10
                2  6
Let the cofactor matrix be C, its determinant is,
   c= | C11   C12 | 
             | C21  C22 |

The cofactor matrix is:
            
The cofactor determinant is,
C= 6  -2 (6)(4)-(-2)(-7)= 24-14=10
We observe that |A| = |C|.
Property of invariance
Let’s suppose a square matrix of order 3:
            
Now, if we add a scalar multiple of one row/column to another to form a new matrix B, the value of the determinant remains unchanged.
Row operation: Ri  Ri, + qRj
Column operation: Ci  Ci + qCj
Where q is any real constant and i  j 
       
       
Then, the determinant of matrix A is equal to the determinant of matrix B; |A| = |B|