Properties of Determinants

Determinants are scalar quantities calculated by adding the sums of the products of the elements of a square matrix according to a set of rules. The determinant aids in the discovery of a matrix adjoint or inverse. We can also use the concept of determinants to solve linear equations using the matrix inversion approach. 

Determinants have various useful qualities. The ten main properties of determinants are:

1. Reflection property 
2. All-zero property
3. Sum property
4. Switching property
5. Scalar multiple properties
6. Invariance property
7. Proportionality or repetition property
8. Triangle property
9. Factor property
10. Cofactor matrix property 

If you want to grasp the concepts better, you must learn the properties of determinants with examples. 

What are the Different Properties of Determinants?

Reflection Property

When the matric columns are transformed to rows and rows are transformed to columns then there’s no change in determinant. Here, both the transpose and the matrix determinant are the same. 

All-zero Property

When all the elements of a column or row are zero, then the value of the determinant is also zero. 

Switching Property

When there are interchanges between any of the two rows or columns then the determinant’s sign changes; however, its absolute value remains the same. This is known as the switching property.

Property of Sum

The determinant is the summation of 2 or more determinants if every constituent is expressed as the summation of at least 2 terms. 

Scalar Multiple Properties

When we multiply a matrix with 2 scalars, we use the two methods: 

• First multiply one scalar by the matrix, find the resulting matrix, then multiply it by another scalar. 
• Multiply the two scalars first, then multiply the result with the matrix.

Invariance Property

When every element of a determinant’s column and row is summed up along with the elements’ equivalents of a different determinant’s column or row, there is no change in the determinant’s value. 

Proportionality/Repetition Property

If a matrix has identical elements in its all columns or rows as the other column or row elements, then the matrix determinant becomes zero. 

Triangle Property

When solving a triangular matrix, its diagonal elements’ product is equal to the determinant. And a matrix determinant will become zero if all elements of the matrix are zero. 

Factor Property

When we multiply every element of a determinant’s column or row with a constant (non-zero), then the same constant is multiplied by the determinant. 

Cofactor Matrix Property

The summation of elements’ product in the determinant of a column or row by the corresponding element’s cofactors of another column or row is null. 

 Characteristics of Determinants

• Determinants are represented similar to a matrix but with a modulus sign.
• The first theorem shows how switching two rows in a matrix affects the determinant. 

det I = 1; 

Let A be an n×n matrix and let B be a matrix which results from switching two rows of A. 

Then det(B) = −det(A).

• If any square matrix B with order n × n has a zero row or a zero column, then det(B) = 0.
• If C is upper-triangular or a lower-triangular matrix, then det(C) is the product of all its diagonal entries.
• If D is a square matrix, then if its row is multiplied by a constant k, then the constant can be taken out of the determinant.

Properties of Determinants II: Some Important Proofs

This section contains several key determinants and cofactor proofs.

First, let’s review what a determinant is. If A = [aij] is an n×n matrix, then detA is defined by calculating the expansions along the first row if A = [aij] is a nxn matrix: 

If  n = 1  then  detA = a1,1 .

             n

detA = ∑a1, icof(A)1,i

           i = 1

In the initial step, we will have to look at whether the given assertion holds true for n = 2 (in the case n = 1 is either completely trivial or meaningless).

After that, we assume the statement is correct for n − 1 (where n ≥ 3 ) and prove it for n. After that, we may deduce that the assertion is true for all n × n matrices for every n ≥ 2, using the Principle of Mathematical Induction. If A is an n × n matrix with 1 ≤ j ≤ n rows, the matrix is generated by eliminating the first column, and A is an n − 1 × n −  1 matrix.

What are the Rules for Operations on Determinants?

To conduct column and row operations on determinants, use the following rules.

• If the columns and rows are swapped, the determinant’s value stays intact.
• If any two types or (two columns) are swapped, the determinant’s sign changes.
• The determinant has 0 value if any two rows or columns of a matrix are equal.
• When each element of a row or column is multiplied by a constant, the determinant’s value is likewise multiplied by the constant.
• The determinant can be stated as a sum of determinants if the row or column elements are expressed as the sum of elements.
• If the elements of a row or column are added or subtracted with the corresponding multiples of elements of another row or column, then the value of the determinant remains unchanged.
• The value of the determinant remains intact when the elements of one row or column are added or subtracted with the matching multiples of elements from another row or column.

Conclusion

Determinants accept a square matrix as input and produce a single integer as output. A square matrix has the same number of rows and columns on both sides. Through determinants, we can find the inverse or adjoint of a matrix. We must also use the notion of determinants to solve linear equations using the matrix inversion approach. The determinant of a matrix, which is the number produced by solving the matrix, was described in this article. Only square matrices of any order are suitable for the determinant.