Properties of Determinants: Triangle, Factor, and Invariance Property

We use properties of determinants to find the determinant of the matrix without expanding it and with lesser calculations. There are various properties of determinants dependent on the operation of elements of rows and columns.

This article aims to provide a detailed understanding of the triangle property, factor property, and property of invariance. Let us learn more about them with examples and FAQs.

What are the Properties of Determinants 

Various properties make the calculation easier for the determinant of the matrix. Some of them are: 

• If we interchange rows and columns, the determinant’s value remains unaffected. 
• If we interchange any two rows or columns, then the sign of the determinant gets changed. 
• If we multiply any row or column with a scalar multiple, the determinant is multiplied by that non-zero number. 
• If any row or column is expressed as a sum of two terms, then the determinant can be split into two determinants.
• If each element of the matrix below or above the main diagonal is zero, then the determinant is equal to the product of diagonals. 
• If each element of a row or column is added with the multiple of another row or element, then the determinant of the matrix remains unchanged.
• Multiplication of determinants of a matrix and its inverse gives 1.

In this study material notes, we will be discussing the three properties of determinants: triangle property, factor property, and property of invariance. 

Triangle Property

If each element of the matrix below or above the main diagonal is 0, then the determinant of the matrix becomes equal to the product of diagonal elements. This is known as the triangle property. 

=a×c×g

=a×e×g

In the first one, the element above the main diagonal is 0; hence, the determinant is equal to the product of the diagonal elements. Similarly, in the second one, the element below the diagonal is zero; hence the determinant becomes the product of the main diagonal. 

For example: 

Δ = 1 × 3 × 3

Δ = 9

Factor Property

As per the factor property, if we have a determinant Δ, and it becomes 0 if we substitute x = p, then (x -p) becomes the factor of the determinant Δ. 

Property of Invariance

Suppose each element of a row or column is added with the multiple of each element of another row or column. In that case, the value of the determinant does not change. This property is known as the property of invariance. Let us prove this with an example. We have been given a matrix of 3 x 3 order as follow:

On expanding the matrix with respect to R1, we get: 

1(2×3 – 1×0) – 0(4×3 – 2×0) +2 (4×1 -2×2)

1 (6 – 0) -0(12) +2(4 – 4)

6 – 0 + 0

Δ = 6

Let us multiply the second row by 2 and add it to the first row. 

R1 -> R1 + R2 + R3

On expanding the matrix along the R1, we get: 

7(2×3 – 1×0) – 3 (4×3 – 2×0) + 5 (4×1 – 2×2)

7 (6) – 3 (12) + 5(4-4)

42 – 36 + 0

Δ= 6

Conclusion

That was all about the properties of determinants. In this study material notes, we learned about the triangle property, property of invariance and factor property. Using these, we can calculate the determinant of the matrix without doing long calculations. 

A determinant is a single numeric value of the matrix. It has various uses not only in mathematics but also in other subjects. Hope this article gives a detailed understanding of the topic.