If the determinant `|(b -c,c -a,a -b),(b' -c',c' -a',a' -b'),(b'' -c'',c'' -a'',a'' - b'')|` is expressible an `m |(a,b,c),(a',b',c'),(a'',b'',c'')|`, then the value of m, is

Options

A. `-1`
B. 0
C. 1
D. 2

Answer

To solve the given problem, we need to evaluate the determinant

\[ D = \begin{vmatrix} b - c & c - a & a - b \\ b' - c' & c' - a' & a' - b' \\ b'' - c'' & c'' - a'' & a'' - b'' \end{vmatrix} \]

and express it in the form \( m \cdot \begin{vmatrix} a & b & c \\ a' & b' & c' \\ a'' & b'' & c'' \end{vmatrix} \).

### Step 1: Simplifying the Determinant

We can simplify the determinant \( D \) by using the property of determinants that allows us to add or subtract columns. Specifically, we will add all three columns together.

\[ D = \begin{vmatrix} (b - c) + (c - a) + (a - b) & c - a & a - b \\ (b' - c') + (c' - a') + (a' - b') & c' - a' & a' - b' \\ (b'' - c'') + (c'' - a'') + (a'' - b'') & c'' - a'' & a'' - b'' \end{vmatrix} \]

### Step 2: Evaluating the First Column

When we add the columns, we observe that:

\[ (b - c) + (c - a) + (a - b) = 0 \]
\[ (b' - c') + (c' - a') + (a' - b') = 0 \]
\[ (b'' - c'') + (c'' - a'') + (a'' - b'') = 0 \]

Thus, the first column becomes a column of zeros:

\[ D = \begin{vmatrix} 0 & c - a & a - b \\ 0 & c' - a' & a' - b' \\ 0 & c'' - a'' & a'' - b'' \end{vmatrix} \]

### Step 3: Determinant of a Matrix with a Zero Column

The determinant of any matrix that has a column of zeros is zero. Therefore, we have:

\[ D = 0 \]

### Step 4: Expressing in Terms of the Second Determinant

We know that \( D \) can also be expressed as:

\[ D = m \cdot \begin{vmatrix} a & b & c \\ a' & b' & c' \\ a'' & b'' & c'' \end{vmatrix} \]

Since we found that \( D = 0 \), we can equate:

\[ 0 = m \cdot \begin{vmatrix} a & b & c \\ a' & b' & c' \\ a'' & b'' & c'' \end{vmatrix} \]

### Step 5: Conclusion

For the equation to hold true, either \( m \) must be zero or the determinant on the right side must be zero. However, since we are looking for a specific value of \( m \), we conclude:

\[ m = 0 \]

Thus, the value of \( m \) is \( 0 \).