Properties of Determinants

A determinant is a function with entries of a square matrix. For a matrix M = [a_ij], the determinant of the matrix is denoted as |M| or det M.

For 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

Various rules and properties are used to easily find the value of the determinant, which are called the Properties of Determinants.

These various properties of the determinant are based on the elements, rows, and columns of the determinant and are discussed 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 & 0\\ d & e & 0\\g & h & i\\\end{pmatrix}

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

Then, according to the Triangle Property of Determinant

|A| = a × e × i

Example:

A = \begin{pmatrix}1 & 0 & 0\\ 2 & 3 & 0\\4 & 5 & 6\\\end{pmatrix}

|A| = \begin{vmatrix}1 & 0 & 0\\ 2 & 3 & 0\\4 & 5 & 6\\\end{vmatrix}

|A| = 1[(3).(6) -(0)(5)] - 0 + 0

= 1[18 - 0] = 18

|A| = 1 × 3 × 6 = 18

Therefore, |A| = a × e × i

2. Determinant of Cofactor Matrix

The determinant of the cofactor matrix of a matrix A is equal to the determinant of A raised to the power of (n-1), where n is the order of the matrix.

For any square matrix A of order n, the property is given by,

|C| = |A| ^n-1

Where,

• |C| - determinant of cofactor matrix
• |A| - determinant of matrix A
• n - order of matrix

Example:

A = \begin{pmatrix}1 & 2 \\ 3 & 4\\\end{pmatrix}

n = 2

|A| = (1)(4) − (2)(3) = 4 − 6 = −2

• C₁₁ = |4| = 4
• C₁₂ = - |3| = −3
• C₂₁ = −|2| = −2
• C₂₂ = |1| = 1

C = \begin{pmatrix}4 & -3 \\ -2 & 1\\\end{pmatrix}

|C| = (4)(1) - (-3)(-2) = 4 - 6 = -2

Therefore |A| = |C|^2-1

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.

Example:

A = \begin{pmatrix}x & 0 \\ 3 & x-2\\\end{pmatrix}

|A| = (x)(x-2) − 0 = x( x - 2)

At x = 2 : 2(x - 2) - 0 = x(x - 2)

(x - a) = (x - 2)

Therefore (x - 2) is a factor of determinant polynomial of x(x - 2)

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 to 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 determinants of matrix A and matrix B are equal, i.e.

|A| = |B|

Example:

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

= (1)(2) - (2)(5) = 2 - 10 = -8

|B| = \begin{vmatrix}1 -3(5) & 2 - 3(2)\\ 5 & 2 \\\end{vmatrix}

= (1-15)(2) - (2-6)(5) = -28 + 20 = -8

Therefore |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 a constant k, the determinant value gets multiplied by k. Constants may be any real number.

det(Δ') = k det(Δ)

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, say |A|'

|A|' = \begin{vmatrix}4 & 2 \\4 & 8 \end{vmatrix}

|A|' = 16 -4 = 12

|A|' = 2 × |A|

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

= (1)(2) - (2)(3) = 2 - 6 = -4

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

= (1)(2) - (3)(2) = 2 - 6 = -4

Therefore |A| = |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}

For a matrix Δ, one row/column exchanged to get Δ', two rows\columns exchanged to get Δ'', then,

det(Δ") = -det(Δ') = det (Δ)

Example:

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

det (Δ) = 1(-35) - 2(-3) - 4(-15) = -35 + 6 + 60 = 31

det (Δ) = 31

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

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

det(Δ') = -4(0 + 15) - 2(3) + 1(35) = -60 -6 + 35 = -31

det (Δ) = -det(Δ')

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

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

det(Δ'') = 7(0 - 5) - 0 -3(-20 - 2) = -35 + 66 = 31

det(Δ'') = det(Δ)

Therefore det(Δ") = -det(Δ') = det (Δ)

8. Repetition Property

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

Example:

A = \begin{pmatrix}1 & 3 \\ 1 & 3 \\\end{pmatrix}

= (1)(3) - (3)(1) = 0

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

Example:

A = \begin{pmatrix}1 & 2 \\ 0 & 0 \\\end{pmatrix}

= (1)(0) - (2)(0) = 0

10. Sum Property

If all the elements of a row or column 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}

Example:

A = \begin{pmatrix} 2+3 & 1\\ 5+8 & 3\\ \end{pmatrix}

|A| = (5)(3) - (1)(13) = 15 - 13 = 2

B = \begin{pmatrix} 2 & 1\\ 5 & 3\\ \end{pmatrix} + \begin{pmatrix} 3 & 1\\ 8 & 3\\\end{pmatrix}

|B| = (6 - 5) + (9 - 8) = 1 + 1 = 2

|A| = |B|

Related Articles

• 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

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}