Find the minor and co-factor of each element of the first column of the following determinants :
`|{:(1,a,bc),(1,b,ca),(1,c,ab):}|`
Hence or otherwise evaluate them.

Answer

To find the minor and cofactor of each element of the first column of the determinant

\[ D = \begin{vmatrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end{vmatrix} \]

we will follow these steps:

### Step 1: Find the Minor of the First Element (1,1)

The minor \( M_{11} \) is obtained by removing the first row and first column:

\[ M_{11} = \begin{vmatrix} b & ca \\ c & ab \end{vmatrix} \]

Calculating this determinant:

\[ M_{11} = b \cdot ab - ca \cdot c = ab^2 - c^2a \]

### Step 2: Find the Minor of the Second Element (2,1)

The minor \( M_{21} \) is obtained by removing the second row and first column:

\[ M_{21} = \begin{vmatrix} a & bc \\ c & ab \end{vmatrix} \]

Calculating this determinant:

\[ M_{21} = a \cdot ab - bc \cdot c = a^2b - b^2c \]

### Step 3: Find the Minor of the Third Element (3,1)

The minor \( M_{31} \) is obtained by removing the third row and first column:

\[ M_{31} = \begin{vmatrix} a & bc \\ b & ca \end{vmatrix} \]

Calculating this determinant:

\[ M_{31} = a \cdot ca - bc \cdot b = ac^2 - b^2c \]

### Step 4: Find the Cofactor of Each Element

The cofactor \( C_{ij} \) is given by the formula:

\[ C_{ij} = (-1)^{i+j} M_{ij} \]

#### Cofactor of the First Element (1,1)

\[ C_{11} = (-1)^{1+1} M_{11} = 1 \cdot (ab^2 - c^2a) = ab^2 - c^2a \]

#### Cofactor of the Second Element (2,1)

\[ C_{21} = (-1)^{2+1} M_{21} = -1 \cdot (a^2b - b^2c) = -a^2b + b^2c \]

#### Cofactor of the Third Element (3,1)

\[ C_{31} = (-1)^{3+1} M_{31} = 1 \cdot (ac^2 - b^2c) = ac^2 - b^2c \]

### Step 5: Evaluate the Determinant

Now we can evaluate the determinant using the cofactors:

\[ D = 1 \cdot C_{11} + 1 \cdot C_{21} + 1 \cdot C_{31} \]

Substituting the values:

\[ D = 1 \cdot (ab^2 - c^2a) + 1 \cdot (-a^2b + b^2c) + 1 \cdot (ac^2 - b^2c) \]

Combining these:

\[ D = ab^2 - c^2a - a^2b + b^2c + ac^2 - b^2c \]

Simplifying:

\[ D = ab^2 - a^2b - c^2a + ac^2 \]

### Final Result

Thus, the final value of the determinant is:

\[ D = ab^2 - a^2b - c^2a + ac^2 \]