20.7 - Property V

In the previous section, we saw the fourth property of determinants. In this section, we will see the fifth property.

Property V
This can be written in 6 steps:
1. Let Δ = $\left |\begin{array}{c}                           
a_1 + \lambda_1    &{    a_2 + \lambda_2    }    &{    a_3 + \lambda_3    }    \\
b_1    &{    b_2    }    &{    b_3    }    \\
c_1    &{    c_2    }    &{    c_3    }    \\
\end{array}\right | $.
2. We can split this determinant and write it as the sum of two determinants:

$\left |\begin{array}{c}                          
a_1 + \lambda_1    &{    a_2 + \lambda_2    }    &{    a_3 + \lambda_3    }    \\
b_1    &{    b_2    }    &{    b_3    }    \\
c_1    &{    c_2    }    &{    c_3    }    \\
\end{array}\right | ~=~\left |\begin{array}{c}                           
a_1     &{    a_2     }    &{    a_3    }    \\
b_1    &{    b_2    }    &{    b_3    }    \\
c_1    &{    c_2    }    &{    c_3    }    \\
\end{array}\right |~+~\left |\begin{array}{c}                           
 \lambda_1    &{ \lambda_2    }    &{ \lambda_3    }    \\
b_1    &{    b_2    }    &{    b_3    }    \\
c_1    &{    c_2    }    &{    c_3    }    \\
\end{array}\right |$

3. This can be verified by evaluating the L.H.S. For that, we will expand along R₁.

4. In the same way, the reader may check the result for other rows and columns.

5. Based on the above steps, we can write:
If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two (or more) determinants.

6. Let us see a simple demonstration prepared using a spreadsheet program:

◼ Remarks:
• 1(magenta color): Here elements of the second column are written as the sum of two numbers. 
• 2(magenta color): Here we apply property V to split the given determinant and write it as the sum of two determinants.
• 3(magenta color): Here each of the three determinants are calculated individually using a spreadsheet program. We see that, L.H.S is equal to the R.H.S.

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Now we have a clear understanding about property V. Let us see an example. It can be written in 2 steps:

1. Show that $\left |\begin{array}{c}                           
a    &{    b    }    &{    c    }    \\
a+2x    &{    b+2y    }    &{    c+2z    }    \\
x    &{    y    }    &{    z    }    \\
\end{array}\right | ~=~0$.
 

2. We can write:

◼ Remarks:
• 1(magenta color): Here we apply property V to split the given determinant and write it as the sum of two determinants.
• 2(magenta color):
    ♦ Consider the first determinant in the R.H.S. The rows R₁ and R₂ are identical. So by property III, the value of this determinant is zero
    ♦ Consider the second determinant in the R.H.S. Every element of R₂ is proportional to the corresponding elements of R₃ by the same ratio 2:1. So by property IV, the value of this determinant is zero.

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In the next section, we will see Property VI.

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