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Properties of Determinants

Properties of determinants are needed to find the value of the determinant with the least calculations. The properties of determinants are based on the elements, the row, and column operations, and it helps to easily find the value of the determinant.

In this article, we will learn more about the properties of determinants and go through some solved examples for a better understanding of the concept.

1.	What Are the Properties of Determinants?
2.	Properties of Determinants
3.	FAQs on Properties of Determinants

What Are the Properties of Determinants?

The properties of determinants are helpful in easily calculating the value of the determinant with simple steps and with the least calculations. The seven important properties of determinants are as follows.

Interchange Property: The value of a determinant remains unchanged if the rows or the columns of a determinant are interchanged.
Sign Property: The sign of the value of determinant changes if any two rows or any two columns are interchanged.
Zero Property: The value of a determinant is equal to zero if any two rows or any two columns have the same elements.
Multiplication Property: The value of the determining becomes k times the earlier value of the determinant if each of the elements of a particular row or column is multiplied with a constant k.
Sum Property: If a few elements of a row or column are expressed as a sum of terms, then the determinant can be expressed as a sum of two or more determinants.
Property Of Invariance: If each element of a row and column of a determinant is added with the equimultiples of the elements of another row or column of a determinant, then the value of the determinant remains unchanged. This can be expressed in the form of a formula as R_i → R_i + kR_j , or C_i → C_i + kC_j.
Triangular Property: If the elements above or below the main diagonal are equal to zero, then the value of the determinant is equal to the product of the elements of the diagonal of the matrix.

Properties of Determinants

Let us check the below seven properties of determinant in detail. The working principle and the formulas, and an explanation of each of the properties are also presented below.

1. Interchange Property

The value of a determinant remains unchanged if the rows or the columns of a determinant are interchanged.

A = \(\begin{vmatrix}a_1&a_2&a_3\\b_1&b_2&b_3\\c_1&c_2&c_3\end{vmatrix}\), A' = \(\begin{vmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{vmatrix}\)

Det(A) = Det(A')

It follows from this property that if the rows and columns of the matrix are interchanged, then the transpose of the matrix is obtained and the determinant value and the determinant of the transpose are equal.

2. Sign Property

The sign of the value of the determinant changes if any two rows or any two columns are interchanged.

A = \(\begin{vmatrix}a_1&a_2&a_3\\b_1&b_2&b_3\\c_1&c_2&c_3\end{vmatrix}\), B = \(\begin{vmatrix}a_1&a_2&a_3\\c_1&c_2&c_3\\b_1&b_2&b_3\end{vmatrix}\)

Det(A) = -Det(B)

The value of the determinant only changes the sign if the row or the column is swapped once. In the above matrix A, the second row has been swapped with the third row to obtain matrix B, and we have Det(A) = -Det(B). If the value of the determinant is D, and the rows or columns are swapped n times, then the new value of the determinant is (-1)ⁿD.

3. Zero Property

The value of a determinant is equal to zero if any two rows or any two columns have the same elements.

A = \(\begin{vmatrix}a_1&a_2&a_3\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}\)

Here the elements of the first row and the second row are identical. Hence the value of the determinant is equal to zero.

Det(A) = 0

4. Multiplication Property

The value of the determining becomes k times the earlier value of the determinant if each of the elements of a particular row or column is multiplied with a constant k.

A = \(\begin{vmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{vmatrix}\), B = \(\begin{vmatrix}ka_1&kb_1&kc_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{vmatrix}\)

Det(B) = k× Det(B)

The elements of the first row are multiplied with a constant k, and the determinant value is also multiplied with the constant k. This property helps in taking a common factor from each row or a column of the determinant. Also if the corresponding elements of any two rows or columns are equal then the value of the determinant is equal to zero.

5. Sum Property

If a few elements of a row or column are expressed as a sum of terms, then the determinant can be expressed as a sum of two or more determinants.

\(\begin{vmatrix}a_1+b_1&a_2 + b_2&a_3+b_3\\c_1&c_2&c_3\\d_1&d_2&d_3\end{vmatrix}\) = \(\begin{vmatrix}a_1&a_2 &a_3\\c_1&c_2&c_3\\d_1&d_2&d_3\end{vmatrix}\) + \(\begin{vmatrix}b_1& b_2&b_3\\c_1&c_2&c_3\\d_1&d_2&d_3\end{vmatrix}\)

The elements of the first row represent the sum of terms, which can be split into two different determinants. Further, the new determinants also have the same second and third row, as the earlier determinant.

6. Property Of Invariance

If each element of a row and column of a determinant is added with the equimultiples of the elements of another row or column of a determinant, then the value of the determinant remains unchanged. This can be expressed in the form of a formula as R_i → R_i + kR_j , or C_i → C_i + kC_j.

A = \(\begin{vmatrix}a_1&a_2&a_3\\b_1&b_2&b_3\\c_1&c_2&c_3\end{vmatrix}\), B = \(\begin{vmatrix}a_1+kc_1&a_2+kc_2&a_3+kc_3\\b_1&b_2&b_3\\c_1&c_2&c_3\end{vmatrix}\)

Det(A) = Det(B)

The elements of the first row of matrix A have been replaced with the sum of the elements of the first row, and the third row multiplied with a constant, to obtain the new matrix B. Here, after this operation also, the determinant A is equal to determinant B.

7. Triangular Property

If the elements above or below the main diagonal are equal to zero, then the value of the determinant is equal to the product of the elements of the diagonal matrix.

\(\begin{vmatrix}a_1&a_2&a_3\\0&b_2&b_3\\0&0&c_3\end{vmatrix}\) = \(\begin{vmatrix}a_1&0&0\\a_2&b_2&0\\a_3&b_3&c_3\end{vmatrix}\) = a₁b₂c₃

Related Articles

The following topics help in a better understanding of the properties of determinants.

Properties of Determinants Examples

Example 1: Find the value of the determinant \(\begin{vmatrix}a&a+b&a+b+c\\2a&3a+2b&4a+3b+2c\\3a&6a+3b&10a+6b+3c\end{vmatrix}\), using the properties of determinants.

Solution:

The given determinant is A = \(\begin{vmatrix}a&a+b&a+b+c\\2a&3a+2b&4a+3b+2c\\3a&6a+3b&10a+6b+3c\end{vmatrix}\).

Here we apply the two operations to the determinants. R₂ → R₂ - 2R₁, and \(R_3 \rightarrow R_3 - 3R_1\)R₃ → R₃ - 3R₁

A = \(\begin{vmatrix}a&a+b&a+b+c\\0&a&2a+b\\0&3a&7a+3b\end{vmatrix}\)

Now we apply the formula R₃ → R₃ - 3R₁

A = \(\begin{vmatrix}a&a+b&a+b+c\\0&a&2a+b\\0&0&a\end{vmatrix}\)

Now expanding along column C₁ we have the following value for the determinant.

|A| = a\(\begin{vmatrix}a&2a+b\\0&a\end{vmatrix}\) + 0 + 0

|A| = a(a² - 0) = a(a²) = a³

Therefore, the value of the determinant is a³.
Example 2: Find the value of the determinant \(\begin{vmatrix}x+y&y+z&z+x\\z&x&y\\1&1&1\end{vmatrix}\), using the properties of determinants.

Solution:

The given determinant is A = \(\begin{vmatrix}x+y&y+z&z+x\\z&x&y\\1&1&1\end{vmatrix}\).

Here we using the operation of \(R_1 \rightarrow R_1 + R_2\) R₁ → R₂₁ + R₂.

A = \(\begin{vmatrix}x+y+z&x+y+z&x+y+z\\z&x&y\\1&1&1\end{vmatrix}\)

Now we can take x + y + z common from the first row.

A = (x + y + z)\(\begin{vmatrix}1&1&1\\z&x&y\\1&1&1\end{vmatrix}\)

Since the elements of the first and the third row are equal, the value of the determinant is zero.

|A| = 0.

Therefore, the value of the given determinant is equal to zero.

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Properties of Determinants Questions

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FAQs on Properties of Determinants

What Are The Properties Of Determinants?

The three important properties of determinants are as follows..

Property 1:The rows or columns of a determinant can be swapped without a change in the value of the determinant.
Property 2: The row or column of a determinant can be multiplied with a constant, or a common factor can be taken from the elements of the row or a column.
Property 3: Two identical rows or columns of a determinant make the value of the determinant equal to zero.

What Are The Uses Of Properties Of Determinants?

The properties of determinants are useful to easily find the value of the determinant with the least calculations. Based on the elements and the row and column operations, the value of the determinant can be easily computed.

How Does The Properties Of Determinants Differ From Properties Of Matrices?

The properties of determinants differed from the properties of matrices, as much as the determinant differs from the matrix. For example, in a determinant, the elements of a particular row or column can be multiplied with a constant, but in a matrix, the multiplication of a matrix with a constant multiplies each element of the matrix.

How to Use Properties of Determinants?

We can use different properties of determinants to solve various mathematical problems depending upon the problem and properties required to solve the problems.

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