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. There are various properties of the determinant of a matrix, and some of the important ones 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?

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 are calculated.

What are Properties of Determinants?

There are various rules and properties that 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 Determinant of a Matrix

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

Triangle Property

This property of the determinant states that if the elements above or below, the main diagonal 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 Traingle Property of Determinant

|A| = a.e.i

Determinant of Cofactor Matrix

For any matrix 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 detrminant of cofactor matrix is,

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

In the above matrix C_ij denotes the cofacts of the respective elements of matrix A.

Factor Property

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

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 the any row or column does not changes the value of determinant, i.e.

R_i → R_i + (q)R_j

OR

C_i → C_i + (q)C_j

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

|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|

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:

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

Δ = 2.1 - 4.3

Δ = 2 - 12 = -10

Now, multiplying by k =  1/2, in first row,

Δ' = 1/2 \begin{vmatrix} 2 & 3\\ 4 & 1 \end{vmatrix}

Δ' = \begin{vmatrix} 2/2 & 3/2\\ 4/2 & 1/2 \end{vmatrix}

Δ' = \begin{vmatrix} 1 & 3/2\\ 2 & 1/2 \end{vmatrix}

Δ' = 1(1) - 4(3/2) = -5

Therefore,

(Δ') = 1/2 (Δ)

det(Δ') = k det(Δ)

Transpose of Determinant (Reflection Property)

Transpose 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 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)

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)

Repetition Property

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

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

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}

Also, Check

• Transpose of a Matrix
• Inverse of a Matrix Formula

Properties of Determinants Examples

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