If `lambda "and" mu` are the cofactors of 3 and -2 respectively in the determinant `|{:(1,0,-2),(3,-1,2),(4,5,6):}|` the value of `lambda+mu` is

Options

A. 5
B. 7
C. 9
D. 11

Answer

To solve the problem, we need to find the cofactors \( \lambda \) and \( \mu \) for the elements 3 and -2 in the given determinant and then compute \( \lambda + \mu \).

The determinant is given as:
\[ D = \begin{vmatrix} 1 & 0 & -2 \\ 3 & -1 & 2 \\ 4 & 5 & 6 \end{vmatrix} \]

### Step 1: Find the cofactor \( \lambda \) of the element 3
The element 3 is located in the first column and the second row (position (2,1)). The cofactor is calculated using the formula:
\[ \text{Cofactor} = (-1)^{m+n} \cdot \text{Minor} \]
where \( m \) is the row number and \( n \) is the column number.

For the element 3:
- \( m = 2 \)
- \( n = 1 \)

So, we have:
\[ \text{Cofactor of 3} = (-1)^{2+1} \cdot \text{Minor of 3} \]
Now, we need to find the minor of 3, which is the determinant of the matrix obtained by removing the second row and first column:
\[ \begin{vmatrix} 0 & -2 \\ 5 & 6 \end{vmatrix} \]

Calculating this determinant:
\[ = (0 \cdot 6) - (-2 \cdot 5) = 0 + 10 = 10 \]
Thus,
\[ \text{Cofactor of 3} = (-1)^{3} \cdot 10 = -10 \]
So, \( \lambda = -10 \).

### Step 2: Find the cofactor \( \mu \) of the element -2
The element -2 is located in the first row and the third column (position (1,3)). Using the same cofactor formula:
\[ \text{Cofactor of -2} = (-1)^{1+3} \cdot \text{Minor of -2} \]
Now, we need to find the minor of -2, which is the determinant of the matrix obtained by removing the first row and third column:
\[ \begin{vmatrix} 1 & 0 \\ 3 & -1 \end{vmatrix} \]

Calculating this determinant:
\[ = (1 \cdot -1) - (0 \cdot 3) = -1 - 0 = -1 \]
Thus,
\[ \text{Cofactor of -2} = (-1)^{4} \cdot (-1) = 1 \]
So, \( \mu = 1 \).

### Step 3: Calculate \( \lambda + \mu \)
Now we can find the value of \( \lambda + \mu \):
\[ \lambda + \mu = -10 + 1 = -9 \]

### Final Answer
The value of \( \lambda + \mu \) is \( -9 \).

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