The determinant of matrix P is denoted as **|P|** i.e. matrix name between two parallel lines. It is also written as **det(P)** or by symbol delta (**Δ**). The determinant is always calculated for a square matrix. So if we talk about matrix in this article then it will be understood as a square matrix. In this article, we will see the **determinant properties** in a lucid manner.

## Properties of determinant

1. If any row of a matrix is completely zero then the determinant of this matrix is zero. For example,

|P| = 0

2. If any column of a matrix is completely zero then the determinant of this matrix is zero. For example,

|A| = 0

3. If any two rows of a matrix are identical then the determinant of this matrix is zero. For example,

|B| = 0

4. If any two columns of a matrix are identical then the determinant of this matrix is zero. For example,

|R| = 0

5. The determinant of a matrix and determinant of a transpose of a matrix is identical. In other words, the determinant of a matrix does not change if rows and columns are interchanged.

|A| = 20

|A^{T}| = 20

6. If two rows of a determinant are interchanged then the value of the determinant is multiplied by (-1). For example,

Now, row 1 and row 3 are interchanged then

7. If two columns of a determinant are interchanged then the value of the determinant is multiplied by (-1). For example,

Now, column 1 and column 2 are interchanged then

8. If P is a square matrix of order n and λ is any scalar then

|λP| = λ^{n }|P|

9. If P and Q be the square matrix of the same order then

|PQ| = |P| |Q|

10. If P be a square matrix then |P^{n}| = (|P|)^{n}

11. If P be a non-singular matrix then

11. If P be a square matrix then

P · Adj(P) = |P|

where Adj(P) is the adjoint of matrix P.

12. |Adj(P)| = |P|^{n-1}

13.

14. If all the elements of any one row are multiplied by a common number (λ) then the determinant becomes λ times the value of the original determinant.

Let multiply all the elements of row 2 of the determinant by 5 then

15. If all the elements of any one column are multiplied by a common number (λ) then the determinant becomes λ times the value of the original determinant.

Let multiply all the elements of column 3 of the determinant by 2 then