# Properties of determinants

The determinants have certain properties that should be known. These properties are very useful to convert the determinants calculation into something a little less slow and tedious.

Let's see some of these properties:

1. Any matrix and its transpose (the transpose matrix is the result of rotating the rows of a matrix to turn them into columns) have the same determinant.

$$\left| A \right| = \left| A^t \right|$$$1. The determinant of a matrix is zero, $$\left| A \right|=0$$, if: 1. The matrix has two equal rows. It is easy to prove this in an exercise for a $$3 \times 3$$ case, for example: $$\begin{matrix} \left| \begin {matrix}a & b & c\\ d & e & f\\ a & b & c \end{matrix}\right| \\ \end{matrix}= a \cdot e \cdot c+ d \cdot b \cdot c + a \cdot b \cdot f- c \cdot e \cdot a - f \cdot b \cdot a - c \cdot b \cdot d =0$$$

1. All the elements of a row are zeros.
2. The elements of a row are a linear combination of other rows. That is:

$$\left| \begin{matrix}2 & 3 & 2 \\ 1 & 2 & 4 \\ 3 & 5 & 6\end{matrix}\right|$$$The 3rd row is a linear combination of the other two ($$f_3=f_1+f_2$$). Without calculating anything, we know that the determinant will be zero. 2. If we swap two parallel rows the determinant changes its sign: $$\left| \begin{matrix}0 & 5 & 1 \\ 1 & 2 & 7 \\ 3 & 1 & 2 \end{matrix}\right|=-\left| \begin{matrix}1 & 2 & 7 \\ 0 & 5 & 1 \\ 3 & 1 & 2\end{matrix}\right|$$$

1. If we add the elements of a row to the elements of a parallel row that have previously been multiplied by a real number, the value of the determinant does not change.

$$\left| \begin{matrix} 1 & 2 & 1 \\ 0 & 2 & 1 \\ 3 & 1 & 2 \end{matrix}\right| \rightarrow C_3= 2\cdot C_1+C_3 \rightarrow \left| \begin{matrix} 1 & 2 & 1 \\ 0 & 2 & 1 \\ 3 & 1 & 2 \end{matrix}\right|=\left| \begin{matrix} 1 & 2 & 3 \\ 0 & 2 & 1 \\ 3 & 1 & 8 \end{matrix}\right|=\left| \begin{matrix} 1 & 2 & 1 \\ 0 & 2 & 1 \\ 3 & 1 & 2 \end{matrix}\right|$$$1. Multiplying a determinant by a real number is the same as multiplying one of its rows by that real number. 2. The determinant of a product is equal to the product of determinants. $$\left| A \cdot B \right|= \left|A\right| \cdot \left| B \right|$$$

Knowing these properties the determinants calculation can be faster. Bearing in mind the 4th property, we can go on modifying our determinant by means of linear combinations in such a way that we can get the largest number of possible $$0$$ or $$1$$, which would reduce the calculations a lot.

$$\left| \begin{matrix} 1 & 3 & 3 & 6 \\ 1 & 3 & 6 & 7 \\ 2 & 4 & 0 & 3 \\ 1 & 5 & 2 & 3 \end{matrix}\right| \rightarrow \begin{array}{c} f_1 \rightarrow f_1 \\ f_2 \rightarrow f_2-f_1 \\ f_3 \rightarrow f_3 -2f_1 \\f_4 \rightarrow f_4-f_1 \end{array} \rightarrow \left| \begin{matrix} 1 & 3 & 3 & 6 \\ 0 & 0 & 3 & 1 \\ 0 & -2 & -6 & -9 \\ 0 & 2 & -1 &-3\end{matrix}\right|$$\$

And as the first column is zero, except for the first element, we will have to calculate the determinant $$\left| \begin{matrix}0 & 3 & 1 \\ -2 & -6 & -9 \\ 2 & -1 & -3\end{matrix}\right|$$ because the other contributions would be zero.