Properties of Determinants – Class 12 Maths

We can see there are 8 important properties of determinants which include Scalar multiple property, Transpose of a determinant (Reflection property) , Row/Column Interchange (Switching Property), Determinant after row operation of adding a proportion of a row to another, Duplicate row determinant (Repetition Property), If all elements of any column or row are zero, then the determinant is zero (All zero property), If all the (sum property) elements of a row or columns in a determinant are expressed as a summation of two or more numbers, then the determinant can be broken down as a sum of corresponding smaller determinants, Determinant of upper triangular or lower triangular matrix (Diagonal Property). All the determinant properties have been covered below in a detailed way along with solved examples.

Scalar Multiple Property

If any row or column of a determinant, is multiplied by any scalar value, that is, a non-zero constant, the entire determinant gets multiplied by the same scalar, that is, if any row or column is multiplied by constant k, the determinant value gets multiplied by k. Constants may be any real number.

Example 1:

Now, on multiplying by k =  1/2, first row,

det(Δ’) = 1(1) – 4(3/2) = -5

Therefore det(Δ’) = 1/2 det(Δ)
⇒ det(Δ’) = k det(Δ)

Example 2:

Now, on multiplying by k =  3  , first column,
                                               2 

Therefore det(Δ’) = 3/2  det(Δ)

⇒ det(Δ’) = k det(Δ)

Transpose of a determinant (Reflection property)

Transpose refers to the operations of interchanging rows and columns of the determinant. The rows become columns and columns become rows in order. It is denoted by |A^T|, for any determinant |A|.
The property says determinant remains unchanged on its transpose, that is, |A^T| = |A|.

Example 1:

⇒ det(A) =  det(A^T)

Example 2:

Row/Column Interchange (Switching Property)

If we interchange any two rows/columns of the determinant, the magnitude (that is, the Sign) changes, but the determinant value remains the same.
Now determinant value = (-1)^{number of exchanges}

Example 1:

If we interchange C₁ and C₃, denoted by C₁↔ C₃

det (Δ) = -det(Δ’)
If we again interchange R₁ and R₂, denoted by R₁↔ R₂

det(Δ”) = -det(Δ’)
             = det (Δ)  

Example 2:

Performing a row operation, we get, 

If we interchange R₁ and R_2, denoted by R₁ ↔ R₂

det(Δ) = – det(Δ’)
If we again interchange R₂ and R₃, denoted by R₂ ↔ R₃

det(Δ”) = – det(Δ’)
            =  det(Δ)

Determinant after row operation of adding a proportion of a row to another

If a multiple of one row is added to another row of determinant, its values remain unchanged.

Example 1:

det (Δ) = 1 + (-26) = -25

R₁ ⇠ R₁ + 2R₂

det (Δ’) = -6 + (-7)(-1) + (-26)
= -6 + 7 – 26  
= -25         
det (Δ) = det (Δ’)

Example 2:

det(Δ) = [8{(5 × 7) – (9 × 3)} – 6{(1 × 7) – (0 × 3)} + 8{(1 × 9) – (5 × 0)}]
det(Δ) = 94                                                                                 

R₃ ⇢ R₃ + 2R₂

det(Δ’) =  [8{(5 × 7) – (9 × 5)} – 6{(1 × 7) – (0 × 5)} + 24{(1 × 9) – (5 × 0)}]
det(Δ’) = 94                                                                                    

det (Δ) = det (Δ’)

Duplicate row determinant (Repetition Property)

If any pair of rows or columns of a determinant are exactly identical or proportion by same amount, then the determinant is zero

Example 1:

det(Δ) = [1{(2 × 0) – (-8 × 3) – 1{(2 × 0) – (-8 × 3)} + 4{(2 × 3) – (2 × 3)}]

det(Δ) = 0 (since R₁ and R₂ are identical)

Example 2:

det(Δ) = [8{(6 × 7) – (6 × 7)} – 5{(1 × 7) – (6 × 0)} + 5{(1 × 7) – (6 × 0)}]

det(Δ) = 0 (since R₂ and R₃ are identical)

All Zero Property

If all elements of any column or row are zero, then the determinant is zero

Example 1:

det(Δ) = 0 (since C₁⇢0)

Example 2:

det(Δ) = 0 (since R₂ ⇢0)

Sum Property

If all the elements of a row or columns in a determinant are expressed as a summation of two or more numbers, then the determinant can be broken down as a sum of corresponding smaller determinants.

Diagonal Property 

If all the elements in the determinant above or below the diagonal are zero, then the determinant is a product of diagonal elements

Example 1:

Example 2:

det(Δ) = 2×1×8×9 = 144