Higher Secondary Mathematics: 20.3

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 Δand 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.

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.