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,
$P=\begin{bmatrix} 1 & 2 & 8\\ 0 & 0 & 0\\ 0 & -9 & 6 \end{bmatrix}$
|P| = 0
2. If any column of a matrix is completely zero then the determinant of this matrix is zero. For example,
$A=\begin{bmatrix} 5 & 1 & 0\\ 6 & 8 & 0\\ 4 & -9 & 0 \end{bmatrix}$
|A| = 0
3. If any two rows of a matrix are identical then the determinant of this matrix is zero. For example,
$B=\begin{bmatrix} 5 & 1 & 2\\ 5 & 1 & 2\\ 9 & 7 & 3 \end{bmatrix}$
|B| = 0
4. If any two columns of a matrix are identical then the determinant of this matrix is zero. For example,
$R=\begin{bmatrix} 8 & 4 & 9\\ 0.3 & 4 & 9\\ 0.56 & 4 & 9 \end{bmatrix}$
|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.
$A=\begin{bmatrix} 1 & 2 & 0\\ -1 & 6 & 1\\ 2 & 0 & 2 \end{bmatrix}$
|A| = 20
$\begin{bmatrix} 1 & -1 & 2\\ 2 & 6 & 0\\ 0 & 1 & 2 \end{bmatrix}$
|AT| = 20
6. If two rows of a determinant are interchanged then the value of the determinant is multiplied by (-1). For example,
$\begin{vmatrix} 1 & 2 & 0\\ -1 & 6 & 1\\ 2 & 0 & 2 \end{vmatrix}=20$
Now, row 1 and row 3 are interchanged then
$\begin{vmatrix} 2 & 0 & 2\\ -1 & 6 & 1\\ 1 & 2 & 0 \end{vmatrix}=(-1)\times 20=-20$
7. If two columns of a determinant are interchanged then the value of the determinant is multiplied by (-1). For example,
$\begin{vmatrix} 1 & 2 & 0\\ -1 & 6 & 1\\ 2 & 0 & 2 \end{vmatrix}=20$
Now, column 1 and column 2 are interchanged then
$\begin{vmatrix} 2 & 1 & 0\\ 6 & -1 & 1\\ 0 & 2 & 2 \end{vmatrix}=(-1)\times 20=-20$
8. If P is a square matrix of order n and λ is any scalar then
|λP| = λn |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 |Pn| = (|P|)n
11. If P be a non-singular matrix then
$\left | P^{-1} \right |=\frac{1}{\left | P \right |}$
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. $\left | Adj(Adj(P)) \right |=\left | P \right |^{(n-1)^{2}}$
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.
$\begin{vmatrix} 1 & 2 & 0\\ -1 & 6 & 1\\ 2 & 0 & 2 \end{vmatrix}=20$
Let multiply all the elements of row 2 of the determinant by 5 then
$\begin{vmatrix} 1 & 2 & 0\\ -5 & 30 & 5\\ 2 & 0 & 2 \end{vmatrix}=5\times 20 = 100$
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.
$\begin{vmatrix} 1 & 2 & 0\\ -1 & 6 & 1\\ 2 & 0 & 2 \end{vmatrix}=20$
Let multiply all the elements of column 3 of the determinant by 2 then
$\begin{vmatrix} 1 & 2 & 0\\ -1 & 6 & 2\\ 2 & 0 & 4 \end{vmatrix}=2\times 20 = 40$
