Properties Of Determinants

We are already familiar with what determinants are. We know that for every square matrix \( [A]_{n×n}\) there exists a determinant to the matrix such that it represents a unique value. In the upcoming discussions, we will learn about certain properties of determinants which simplify the calculation of the determinant.

Properties Of Determinants

1. Property of Reflection:

If in a determinant, even when rows are interchanged with columns, the value of the determinant remains unaltered.

If \( \triangle= \begin{vmatrix}a & b & c\cr d & e & f \cr g & h & i\end{vmatrix}~ and ~\triangle_1= \begin{vmatrix}a & d & g\cr b & e & h\cr c & f & i \end{vmatrix} \)

In such a case \(\triangle\) = \(\triangle_1 \)

Note: Interchange of rows is denoted as\( R_i ↔ R_j \) and the column interchange is denoted as \( C_i ↔ C_j \)

2.  All Zero Property:

If all the elements of any particular row or any particular column are zero then the value of the determinant is also zero.

Example: Evaluate the determinant \( \triangle =\begin{vmatrix}
0 & 155 & 38\cr 0 & 309 & -56\cr 0 & 456 & -937 \end{vmatrix} \)
.

Solution: As the column is entirely zero, therefore the value of the determinant is zero, by the all-zero property.

3.  Repetition Property:

If there are two identical rows or columns then the value of the determinant is zero.

Example: Evaluate the determinant \( \triangle =\begin{vmatrix} 155 & 155 & 38\cr 309 & 309 & -56\cr 456 & 456 & -937 \end{vmatrix}\).

Solution: As the column 1 and column 2 are exactly identical therefore the value of the determinant is zero, by repetition property.

4. Switching Property:

If any two rows or columns are interchanged then the value of the determinant changes its sign i.e. If \( \triangle =\begin{vmatrix} a & b & c\cr d & e & f\cr g & h & i \end{vmatrix}~and~ \triangle_1 =\begin{vmatrix} c & b & a\cr f & e & d\cr i & h & g \end{vmatrix}\).

Then by the property of switching \( \triangle = -\triangle_1\)

5. Property of multiplication by a Scalar:

If all the elements of a row or a column in a determinant are multiplied by a constant non-zero value then the value of the determinant also becomes a multiple of that constant i.e. If \(\triangle =\begin{vmatrix} a & kb & c\cr d & ke & f\cr g & kh & i \end{vmatrix}\).

Then from the property of scalar multiplication it follows that

\(\triangle = k\begin{vmatrix} a & b & c\cr d & e & f\cr g & h & i \end{vmatrix}\).

6. Property of Sum:

If \( =\begin{vmatrix} a+x & b & c\cr d+y & e & f\cr g+z & h & i \end{vmatrix}\).

Then the value of determinant becomes equal to the sum of the determinants obtained by splitting, i.e. \( \triangle =\begin{vmatrix} a+x & b & c\cr d+y & e & f\cr g+z & h & i \end{vmatrix} =\begin{vmatrix} x & b & c\cr y & e & f\cr z & h & i \end{vmatrix}+\begin{vmatrix} a & b & c\cr d & e & f\cr g & h & i \end{vmatrix} \).

7. Property of Invariance:

If any row or column of a determinant is expressed as the sum of multiples of any other row or column then the value of the determinant remains unchanged.

\( \triangle = \begin{vmatrix} a & b & c\cr d & e & f\cr g & h & i \end{vmatrix} =\begin{vmatrix} a+ \alpha b+\beta c & b & c\cr d + \alpha e + \beta f & e & f\cr g + \alpha h + \beta i & h & i \end{vmatrix}\).

8. Triangle Property:

If, in a determinant all the elements above or below the diagonal consists of zero, only then the value of the determinant will be equal to the product of the diagonal element.

\( \triangle =\begin{vmatrix} a & b & c\cr 0 & e & f\cr 0 & 0 & i \end{vmatrix} =\begin{vmatrix} a & 0 & 0\cr d & e & 0\cr g & h & i \end{vmatrix} \) = a × e × i.

There you go!! These are the basic properties of determinants and we are sure you want to learn more. To know more about different concepts like matrices and determinants across different subjects download BYJU’S-The Learning App.


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In the below figure,

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