The minor `M_(31)` if ` Delta = |(1, a, bc),(1, b, ca),(1, c, ab)|` is

Options

A. ` - c ( a^(2) - b^(2))`
B. ` x ( b^(2) - a^(2))`
C. ` c( a^(2) + b^(2))`
D. ` c ( a^(2) - b^(2))`

Answer

To find the minor \( M_{31} \) of the determinant \( \Delta = \begin{vmatrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end{vmatrix} \), we will follow these steps:

### Step 1: Identify the Minor
The minor \( M_{31} \) is the determinant formed by removing the 3rd row and 1st column from the original determinant \( \Delta \).

### Step 2: Form the Reduced Matrix
After removing the 3rd row and 1st column, we are left with the following 2x2 matrix:
\[ \begin{vmatrix} a & bc \\ b & ca \end{vmatrix} \]

### Step 3: Calculate the Determinant of the 2x2 Matrix
The determinant of a 2x2 matrix \( \begin{vmatrix} p & q \\ r & s \end{vmatrix} \) is calculated as \( ps - qr \). Therefore, for our matrix:
\[ M_{31} = (a)(ca) - (bc)(b) = aca - b^2c \]

### Step 4: Factor the Result
Now, we can factor out \( c \) from the expression:
\[ M_{31} = c(a^2 - b^2) \]

### Final Answer
Thus, the minor \( M_{31} \) is:
\[ M_{31} = c(a^2 - b^2) \]

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