20.5 - Property III

In the previous section, we saw the second property of determinants. In this section, we will see the third property.

Property III
This can be written in 9 steps:
1. Let Δ = $\left |\begin{array}{r}                           
a_1    &{    a_2    }    &{    a_3    }    \\
b_1    &{    b_2    }    &{    b_3    }    \\
a_1    &{    a_2    }    &{    a_3    }    \\
\end{array}\right | $.
• We see that, R₁ and R₃ are identical.
2. Let us evaluate Δ. We will expand along R₁.

 

◼ Remarks:
• In 3(magenta color),
    ♦ First term = -1 × sixth term 
    ♦ Second term = -1 × fourth term 
    ♦ Third term = -1 × fifth term

• So the sum is zero. 

3. In the same way, the reader may check the result when any two columns are identical. It will be zero.

4. Based on the above steps, we can write:
If any two rows (or columns) of a determinant are identical, then the value of that determinant will be zero.

5. The proof can be written in 6 steps:
(i) Suppose that, two rows of a matrix A are identical.
(ii) Calculate det(A)
(iii) Interchange the identical rows and write a new matrix A₁.
(iii) Calculate det(A₁)
(iv) Applying property II, we can write: -det(A) = det(A₁)
(v) In the matrix A, we interchanged identical rows.
So A = A₁
(vi) Since A = A₁, the determinants will also be equal.
That is., det(A) = det(A₁)
(vii) Substituting the above value of det(A₁) into (iv), we get:
-det(A) = det(A)
• This is possible only if det(A) = 0

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Now we have a clear understanding about property III. Let us see an example. It can be written in 5 steps:

1. Let Δ = $\left |\begin{array}{r}                           
5    &{    6    }    &{    5    }    \\
-4    &{    3    }    &{   -4    }    \\
1    &{    9    }    &{    1    }    \\
\end{array}\right | $.
2. Let us evaluate Δ. We will expand along R₁.

4. In this example, C₁ and C₃ are identical.

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In the next section, we will see Property IV.

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20.4 - Property II

In the previous section, we saw the first property of determinants. In this section, we will see the second property.

Property II
This can be written in 9 steps:
1. Let Δ = $\left |\begin{array}{r}                           
a_1    &{    a_2    }    &{    a_3    }    \\
b_1    &{    b_2    }    &{    b_3    }    \\
c_1    &{    c_2    }    &{    c_3    }    \\
\end{array}\right | $.
2. We can write a new determinant Δ₁ by interchanging any two rows (or columns). For example, by interchanging the second and third rows, we get:
Δ₁ = $\left |\begin{array}{r}                           
a_1    &{    a_2    }    &{    a_3    }    \\
c_1    &{    c_2    }    &{    c_3    }    \\
b_1    &{    b_2    }    &{    b_3    }    \\
\end{array}\right |$.
3. Let us evaluate Δ. We will expand along R₁.

4. Let us evaluate Δ₁. We will expand along R₁.

5. Let us compare Δ and Δ₁. Both are written together below:

• Identical terms are given the same number. It is easy to see that, all six terms are identical. But the signs are opposite.

6. Let us compare (−Δ) and Δ₁. Both are written together below:

• Identical terms are given the same number. It is easy to see that, all six terms are identical. The signs are also identical.
• So we get: −Δ = Δ₁.

7. In the same way, the reader may check the result by interchanging any two columns.

8. Based on the above steps, we can write:
If any two rows (or columns) of a determinant are interchanged, then the sign of the determinant changes.

9. Suppose that:
    ♦ R_i and R_j represent the i^th and j^th rows respectively.
    ♦ C_i and C_j represent the i^th and j^th columns respectively.

• Then the process of interchanging the two rows can be represented as R_i ↔ R_j.
• Also, the process of interchanging the two columns can be represented as C_i ↔ C_j.

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Now we have a clear understanding about property II. Let us see an example. It can be written in 5 steps:

1. Let Δ = $\left |\begin{array}{r}                           
2    &{    -3    }    &{    5    }    \\
6    &{    0    }    &{    4    }    \\
1    &{    5    }    &{    7    }    \\
\end{array}\right | $.
2. Let us do C₁ ↔ C₃. We get:
Δ₁ = $\left |\begin{array}{r}                           
5    &{    -3    }    &{    2    }    \\
4    &{    0    }    &{    6    }    \\
7    &{    5    }    &{    1    }    \\
\end{array}\right |$.
3. Let us evaluate Δ. We will expand along R₂.

4. Let us evaluate Δ₁. We will expand along C₂.

5. Comparing the results in (3) and (4), we see that:
−Δ = Δ₁. 

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In the next section, we will see Property III.

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20.3 - Property I

In the previous section, we saw how to obtain the determinant of order 3. In this section, we will see some properties of determinants.

• The properties will help us to obtain maximum number of zeroes in a row/column.
• When zeroes are obtained in this way, the evaluation of the determinant will become easier.
• The properties are applicable to determinants of any order. But for our present discussion, we will consider order 3 only.

Property I
This can be written in 8 steps:
1. Let Δ = $\left |\begin{array}{r}                           
a_1    &{    a_2    }    &{    a_3    }    \\
b_1    &{    b_2    }    &{    b_3    }    \\
c_1    &{    c_2    }    &{    c_3    }    \\
\end{array}\right | $.
2. We can write a new determinant Δ₁ by interchanging rows and columns. That is.,
Δ₁ = $\left |\begin{array}{r}                           
a_1    &{    b_1    }    &{    c_1    }    \\
a_2    &{    b_2    }    &{    c_2    }    \\
a_3    &{    b_3    }    &{    c_3    }    \\
\end{array}\right |$.
3. Let us evaluate Δ. We will expand along R₁.

4. Let us evaluate Δ₁. We will expand along R₁.

5. Let us compare Δ and Δ₁. Both are written together below:

• Identical terms are given the same number. It is easy to see that, all six terms are identical. The signs are also identical.
• So we get: Δ = Δ₁.

6. Based on the above steps, we can write:
The determinant remains unchanged if it's rows and columns are interchanged.

7. Recall that, transpose of a matrix is obtained by interchanging rows and columns of that matrix. So we can write:
If A is a square matrix, then det(A) = det(A')

8. If R_i and C_i represent the i^th row and i^th column respectively, then the process of interchanging the rows and columns can be represented as R_i ↔ C_i.

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Now we have a clear understanding about property I. Let us see an example. It can be written in 5 steps:

1. Let Δ = $\left |\begin{array}{r}                           
2    &{    -3    }    &{    5    }    \\
6    &{    0    }    &{    4    }    \\
1    &{    5    }    &{    7    }    \\
\end{array}\right | $.
2. By interchanging rows and columns, we get:
Δ₁ = $\left |\begin{array}{r}                           
2    &{    6    }    &{    1    }    \\
-3    &{   0    }    &{    5    }    \\
5    &{    4    }    &{    7    }    \\
\end{array}\right |$.
3. Let us evaluate Δ. We will expand along R₂.

4. Let us evaluate Δ₁. We will expand along R₂.

5. Comparing the results in (3) and (4), we see that:
Δ = Δ₁. 

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In the next section, we will see Property II.

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