Properties of Determinants

Properties of Determinants are the properties that are required to solve various problems in Matrices. There are various properties of the determinant that are based on the elements, rows, and columns of the determinant. These properties help us to easily find the value of the determinant. Suppose we have a matrix M = [a_ij] then the determinant of the matrix is denoted as, |M| or det M.

Some of the important properties of the determinant of matrices are, Reflection Property, Switching Property, Scalar Multiple Properties, Sum Property, Invariance Property, Factor Property, Triangle Property, Co-Factor Matrix Property, All-Zero Property, and Proportionality or Repetition Property.

In this article, we will learn about all the properties of the determinant with examples and others in detail.

What are Determinants and their Properties?

The determinant of a matrix is the value obtained by solving any square matrix in a particular order. Suppose we have a square matrix A of order 2, i.e.

A = \begin{pmatrix}a & b\\ c & d\end{pmatrix}

Then determinant of A is defined as,

|A| = \begin{vmatrix}a & b\\ c & d\end{vmatrix}

|A| = ad - bc

This is the value of the determinant of matrix A, similarly, the determinant of all square matrices is calculated.

Also Read: Determinant of a Matrix

Properties of Determinants

Some various rules and properties are used to easily find the value of the determinant which are called the Properties of Determinants. There are various properties of the determinant and some of the important ones are,

• Triangle Property
• Determinant of Cofactor Matrix
• Factor Property
• Property of Invariance
• Scalar Multiple Property
• Reflection Property
• Switching Property
• Repetition Property
• All Zero Property
• Sum Property

Now let's learn about them in detail.

Properties of Determinants of a Matrix Explained

The various properties of determinants of a Matrix are discussed in detail below:

1. Triangle Property

This property of the determinant states that if the elements above or below the main diagonal are zero, then the value of the determinant is equal to the product of the diagonal elements.

For any square matrix A such that,

A = \begin{pmatrix}a & 0 & o\\ d & e & 0\\g & h & i\\\end{pmatrix}

|A| = \begin{vmatrix}a & 0 & o\\ d & e & 0\\g & h & i\\\end{vmatrix}

Then according to Triangle Property of Determinant

|A| = a*e*i

2. Determinant of Cofactor Matrix

For any square matrix A of order 2,

A = \begin{pmatrix}a & b\\ c & d\end{pmatrix}

Then determinant of A is defined as,

|A| = \begin{vmatrix}a & b\\ c & d\end{vmatrix}

Now suppose the cofactor matrix is, C then the determinant of the cofactor matrix is,

|C| = \begin{vmatrix}C_{11} & C_{12}\\ C_{21} & C_{22}\end{vmatrix}

In the above matrix, C_ij denotes the cofactors of the respective elements of matrix A.

3. Factor Property

For any square matrix A of variable 'x' if on putting x = a the value of the determinant is zero then, (x - a) is a factor of the determinant.

4. Property of Invariance

Suppose we have a square matrix A of order 3

A = \begin{pmatrix}a & b & c\\ d & e & f\\g & h & i\\\end{pmatrix}

Then adding a scalar multiple of any row or column with any row or column does not change the value of the determinant, i.e.

R_i → R_i + (q)R_j
OR
C_i → C_i + (q)C_j

where q represents the scalar constant, then the value of the determinant of the new matrix form does not change.

|A| = \begin{vmatrix}a & b & c\\ d & e & f\\g & h & i\\\end{vmatrix}

|B| = \begin{vmatrix}a + qc & b & c\\ d + qf& e & f\\g + qi& h & i\\\end{vmatrix}

Then the determinant of matrix A and matrix B are equal, i.e.

|A| = |B|

5. Scalar Multiple Property

If any row or column of a determinant, is multiplied by any scalar value, that is a non-zero constant, the entire determinant gets multiplied by the same scalar, that is, if any row or column is multiplied by constant k, the determinant value gets multiplied by k. Constants may be any real number.

Example:

\begin{vmatrix}2 & 1\\ 2 & 4\\ \end{vmatrix}

Its determinant is |A| = 8 - 2 = 6

Let us multiply all the elements in the above matrix by 2.

\begin{vmatrix}4 & 2 \\2 & 4 \end{vmatrix}

|A| = 16 -4 = 12

You can see in the above example that after multiplying one row by a number 2, the determinant of the new matrix was also multiplied by the same number 2.

Let's check for the correctness:

Original determinant = -10 and the new determinant = -5/2

So, (-5/2)/(-10) = 5/20 = 1/4

Again, (1/4) (-10) simplifies to -5/2.

Hence, the new determinant (-5/2) is equal to the original determinant (-10) multiplied by the scalar 1/4, which confirms the scalar multiple property.

=> New determinant=Original determinant × k

Therefore,

(Δ') = 1/4 (Δ)

det(Δ') = k det(Δ)

6. Transpose of Determinant (Reflection Property)

Transpose of a Matrix refers to the operations of interchanging rows and columns of the determinant. The rows become columns and columns become rows in order. It is denoted by |A^T|, for any determinant |A|.
The property says the determinant remains unchanged on its transpose, that is, |A^T| = |A|.

Example:

|A| = \begin{vmatrix} 0 & 1 & 2\\ 3 & 4 & 5\\1&0&5\end{vmatrix}

|A| = -12

|A^T| = \begin{vmatrix} 0 & 3 & 1\\ 1 & 4 & 0\\2 & 5 & 5\end{vmatrix}

|A^T| = -12

det(A) =  det(A^T)

7. Switching Property

If we interchange any two rows/columns of the determinant, the magnitude (i.e. the sign) changes, but the determinant value remains the same.

Now,

Value of Determinant = (-1)^{number of exchanges}

Example: Apply switching property in,

Δ = \begin{vmatrix} 1 & 2 & -4\\ -3 & 0 & 7\\0 & 5 & 1\end{vmatrix}

If we interchange C₁ and C₃, denoted by C₁ ↔ C₃

Δ' = \begin{vmatrix} -4 & 2 & 1\\ 7 & 0 & -3\\1 & 5 & 0\end{vmatrix}

det (Δ) = -det(Δ')

If we again interchange R₁ and R₂, denoted by R₁ ↔ R₂ 

Δ'' = \begin{vmatrix} 7 & 0 & -3\\ -4 & 2 & 1\\1 & 5 & 0\end{vmatrix}

det(Δ") = -det(Δ') = det (Δ)

8. Repetition Property

If any pair of rows or columns of a determinant are identical or proportioned by the same amount, then the determinant is zero.

9. All Zero Property

If all elements of any column or row are zero, then the determinant is zero

For any matrix,

A = \begin{pmatrix}0 & 0 & 0\\ d & e & g\\g & h & i\\\end{pmatrix}

⇒ |A| = 0

10. Sum Property

If all the elements of a row or columns in a determinant are expressed as a summation of two or more numbers, then the determinant can be broken down as a sum of corresponding smaller determinants.

We have, \Delta = \begin{vmatrix} a + 5m & d & g\\ b + 7n & e & h\\ c + 3p & f & i \end{vmatrix}  

Then, \Delta = \begin{vmatrix} a & d & g\\ b & e & h\\ c & f & i \end{vmatrix} + \begin{vmatrix} 5m & d & g\\ 7n & e & h\\ 3p & f & i \end{vmatrix}

People Also Read:

• Transpose of a Matrix
• Inverse of a Matrix Formula
• Determinant of 2x2 Matrix
• Determinant of 3 × 3 Matrix

Solved Examples of Properties of Determinants

Example 1: Verify det(Δ') = k det(Δ) in,

Δ = \begin{vmatrix} 5 & 2 & 3\\ 2 & 4 & 5\\ 1 & 8 & 7 \end{vmatrix} k = 3/2

Solution:

Δ = \begin{vmatrix} 5 & 2 & 3\\ 2 & 4 & 5\\ 1 & 8 & 7 \end{vmatrix}
det(Δ) = 5[(4×7) - (8×5)] - 2[(2×7) - (5×1)] + 3[(2×8) - (4×1)]
det(Δ) = -60 - 18 + 36
det(Δ) = -42

Now, on multiplying by k = 3/2, first column,

Δ' = \begin{vmatrix} 5 & 2 & 3\\ 2 & 4 & 5\\ 1 & 8 & 7 \end{vmatrix}
Δ' = \begin{vmatrix} 15/2 & 2 & 3\\ 3 & 4 & 5\\ 3/2 & 8 & 7 \end{vmatrix}
det (Δ') = -63

Therefore,
det(Δ') = 3/2  det(Δ)

⇒ det(Δ') = k det(Δ)

Example 2: Find the Determinant of

[A] = \begin{pmatrix}9 & 8 & 7\\ 0 & 0 & 0\\1 & 8 & 5\\\end{pmatrix}

Solution:

|A| = \begin{vmatrix}9 & 8 & 7\\ 0 & 0 & 0\\1 & 8 & 5\\\end{vmatrix}
det(Δ) = 0 (since R₂ ⇢0)

Example 3: Find the Determinant of

[A] = \begin{pmatrix}2 & 0 & 0 & 0\\ 3 & 1 & 0 & 0\\5 & 6 & 8 & 0\\7 & 1 & 5 & 9\end{pmatrix}

Solution:

|A| = \begin{vmatrix}2 & 0 & 0 & 0\\ 3 & 1 & 0 & 0\\5 & 6 & 8 & 0\\7 & 1 & 5 & 9\end{vmatrix}
det(Δ) = 2 × 1 × 8 × 9
det(Δ) = 144

Properties of Determinants Class 12

In a Class 12 Mathematics curriculum, the properties of determinants are typically covered as part of linear algebra. Understanding and applying these properties of determinants enable students to efficiently manipulate matrices and solve problems in linear algebra.

Also Check:

• Determinants Class 12 Notes
• Determinants Class 12 NCERT Solutions

Practice Problems on Properties of Determinants

Question 1: Calculate the determinant of the following matrix and state how row swapping affects the determinant:

\begin{vmatrix}2 & 3 \\1 & 4\end{vmatrix}

Swap the rows and then calculate the determinant again.

Question 2: Determine the determinant of the matrix below. Then, multiply the first row by 3 and find the new determinant:

\begin{vmatrix}1 & -1 \\2 & 3\end{vmatrix}

Compare the original determinant with the new one to explain the effect of scalar multiplication on the determinant.

Question 3: Calculate the determinant before and after performing a row operation where you add twice the first row to the second row:

\begin{vmatrix}1 & 2 \\3 & 4\end{vmatrix}

Question 4: Calculate the determinant of the following triangular matrix:

\begin{vmatrix}5 & 0 & 0 \\-1 & 3 & 0 \\2 & -2 & 4\end{vmatrix}

Discuss why the determinant of a triangular matrix is the product of its diagonal elements.

Question 5: Find the determinant of the matrix below, noting what happens when rows are proportional:

\begin{vmatrix}3 & 6 \\6 & 12\end{vmatrix}