Question

If the cofactors of the elements 3, 7, and 6 of the matrix \( \begin{bmatrix} 1 & 2 & 3 \\ 4 & -1 & 7 \\ 2 & 4 & 6 \end{bmatrix} \) are \( a, b, c \) respectively, then evaluate: \[ \left[ a \ b \ c \right] \begin{bmatrix} 1 \\ 4 \\ 2 \end{bmatrix} + \left[ a \ b \ c \right] \begin{bmatrix} 3 \\ 7 \\ 6 \end{bmatrix} \]

• A. \( -1 \)
• B. \( 1 \)
• C. \( 0 \)
• D. \( 3 \)

Hint:

When summing dot products, it’s efficient to combine vectors first, then apply cofactors. Pay attention to signs in cofactor expansion.

Answer 1

The Correct Option is C

We are given matrix: \[ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & -1 & 7 \\ 2 & 4 & 6 \end{bmatrix} \] and the cofactors of entries 3, 7, and 6, which are the elements \( A_{13}, A_{23}, A_{33} \) respectively. We define: \[ [a\ b\ c] = \text{cofactors of } (3, 7, 6) \] We need to evaluate: \[ [a\ b\ c] \cdot \left( \begin{bmatrix} 1 \\ 4 \\ 2 \end{bmatrix} + \begin{bmatrix} 3 \\ 7 \\ 6 \end{bmatrix} \right) = [a\ b\ c] \cdot \begin{bmatrix} 4 \\ 11 \\ 8 \end{bmatrix} \] Now calculate each cofactor:

• \( a = \text{Cofactor of } 3 = (-1)^{1+3} \cdot \begin{vmatrix} 4 & -1 \\ 2 & 4 \end{vmatrix} = 1 \cdot (16 + 2) = 18 \)
• \( b = \text{Cofactor of } 7 = (-1)^{2+3} \cdot \begin{vmatrix} 1 & 2 \\ 2 & 4 \end{vmatrix} = -1 \cdot (4 - 4) = 0 \)
• \( c = \text{Cofactor of } 6 = (-1)^{3+3} \cdot \begin{vmatrix} 1 & 2 \\ 4 & -1 \end{vmatrix} = 1 \cdot (-1 - 8) = -9 \)

So, \( [a\ b\ c] = [18\ 0\ -9] \) Now compute: \[ 18 \cdot 4 + 0 \cdot 11 + (-9) \cdot 8 = 72 + 0 - 72 = 0 \] Final Answer: \[ \boxed{0} \]

Answer 2

Step 1: Identify the elements and their cofactors.
The given matrix is:
\[ M = \begin{bmatrix} 1 & 2 & 3 \\ 4 & -1 & 7 \\ 2 & 4 & 6 \end{bmatrix}. \]
The elements whose cofactors are \( a, b, c \) are:
- \(3\) at position \( (1,3) \),
- \(7\) at position \( (2,3) \),
- \(6\) at position \( (3,3) \).

Step 2: Recall the properties of cofactors.
The cofactor matrix \( C \) is the matrix of cofactors, and the adjoint matrix is the transpose of \( C \).
Here, \( a, b, c \) are cofactors corresponding to the elements in the third column.

Step 3: Express the problem in terms of cofactors and vectors.
We need to evaluate:
\[ \left[ a \quad b \quad c \right] \begin{bmatrix} 1 \\ 4 \\ 2 \end{bmatrix} + \left[ a \quad b \quad c \right] \begin{bmatrix} 3 \\ 7 \\ 6 \end{bmatrix}. \]
This simplifies to:
\[ \left[ a \quad b \quad c \right] \left( \begin{bmatrix} 1 \\ 4 \\ 2 \end{bmatrix} + \begin{bmatrix} 3 \\ 7 \\ 6 \end{bmatrix} \right) = \left[ a \quad b \quad c \right] \begin{bmatrix} 4 \\ 11 \\ 8 \end{bmatrix}. \]

Step 4: Interpret the vector of cofactors \( [a \ b \ c] \).
The vector \( [a \ b \ c] \) corresponds to the cofactors of the third column of \( M \), which is the third row of the adjoint matrix \( \text{adj}(M) \) transposed.
Since the adjoint matrix satisfies:
\[ M \cdot \text{adj}(M) = (\det M) I, \] and the third column of \( \text{adj}(M) \) contains cofactors of the third column of \( M \).

Step 5: Use the property of matrix and adjoint.
Multiplying matrix \( M \) by the third column of \( \text{adj}(M) \) gives:
\[ M \begin{bmatrix} a \\ b \\ c \end{bmatrix} = (\det M) \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}. \]
Because the product selects the third column of the identity matrix.

Step 6: Multiply \( [a \ b \ c] \) with the sum vector.
We can rewrite:
\[ \left[ a \quad b \quad c \right] \begin{bmatrix} 4 \\ 11 \\ 8 \end{bmatrix} = \begin{bmatrix} a & b & c \end{bmatrix} \cdot \begin{bmatrix} 4 \\ 11 \\ 8 \end{bmatrix}. \]

From Step 5:
\[ M \begin{bmatrix} a \\ b \\ c \end{bmatrix} = (\det M) \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}. \]
Multiply both sides by the vector \( \begin{bmatrix} 1 & 4 & 2 \end{bmatrix} \) from the left:
\[ \begin{bmatrix} 1 & 4 & 2 \end{bmatrix} M \begin{bmatrix} a \\ b \\ c \end{bmatrix} = (\det M) \begin{bmatrix} 1 & 4 & 2 \end{bmatrix} \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} = (\det M) \times 2. \]

Since this value is related to the expression, but the sum simplifies directly to zero because the cofactors \( a, b, c \) satisfy the properties of cofactors and determinants such that:
\[ [a \quad b \quad c] \cdot \begin{bmatrix} 1 \\ 4 \\ 2 \end{bmatrix} = - [a \quad b \quad c] \cdot \begin{bmatrix} 3 \\ 7 \\ 6 \end{bmatrix}. \]

Step 7: Final evaluation.
Therefore:
\[ \left[ a \quad b \quad c \right] \begin{bmatrix} 1 \\ 4 \\ 2 \end{bmatrix} + \left[ a \quad b \quad c \right] \begin{bmatrix} 3 \\ 7 \\ 6 \end{bmatrix} = 0. \]