Question

What are the Properties of determinants?

Answer 1

Properties of determinants.

There are $10$ main properties of determinants which are listed below.

Reflection property:

The determinant remains unchanged if its rows are changed into columns and the columns into rows. This is known as the property of reflection.

All zero property:

If all the elements of a row (or column) are zero, then the determinant is zero.

Proportionality (Repetition) Property:

If all elements of a row (or column) are proportional (identical) to the elements of some other row (or column), then the determinant is zero.

Switching Property:

The interchange of any two rows (or columns) of the determinant changes its sign.

Scalar Multiple Property:

If all the elements of a row (or column) of a determinant are multiplied by a non-zero constant, then the determinant gets multiplied by the same constant.

Sum property:

$\left|\begin{array}{ccc}a+m& b& c\\ d+n& e& f\\ g+o& h& i\end{array}\right|=\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|+\left|\begin{array}{ccc}m& b& c\\ n& e& f\\ o& h& i\end{array}\right|$

Property of Invariance:

$\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|=\left|\begin{array}{ccc}{a}_1+\alpha b_1+\beta c_1& b_1& c_1\\ a_2+\alpha b_2+\beta c_2& b_2& c_2\\ a_3+\alpha b_3+\beta c_3& b_3& c_3\end{array}\right|$

A determinant remains unchanged under an operation of the form $C_i\to C_i+\alpha C_j+\beta C_k$ , where $j,k\ne i$

or an operation has form $R_i\to R_i+\alpha R_j+\beta R_k$ , where $j,k\ne i$.

Factor Property:

If a determinant becomes zero when we put $x=\alpha$. Then, $(x-\alpha )$ is a factor of the determinant.

Triangle Property:

If all the elements of a determinant above or below the main diagonal consist of zeros, then the determinant is equal to the product of diagonal elements.

$\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ 0& b_2& c_2\\ 0& 0& c_3\end{array}\right|=\left|\begin{array}{ccc}{a}_1& 0& 0\\ a_2& b_2& 0\\ a_3& b_3& c_3\end{array}\right|=a_1{b}_2{c}_3$

Determinant of cofactor matrix:

$∆=\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|$ then ${∆}_1=\left|\begin{array}{ccc}{C}_{11}& C_{12}& C_{13}\\ C_{21}& C_{22}& C_{23}\\ C_{31}& C_{32}& C_{33}\end{array}\right|$ , where $C_{ij}$ denotes the cofactor of the element $a_{ij}$ in $∆$.