A is ` 2 xx 2 ` matrix whose elements are given by `a_(ij) = 2 I_(ij) - j . ` Then find `M_(21) +A_(11)` , where ` M_(ij) ` is the minor of `a_(ij)` and `A _(ij) ` is the co-factor of ` a_(ij)`.

Options

A. a) 0
B. b) -2
C. c) 4
D. d) 1

Answer

To solve the problem, we need to find the values of \( M_{21} \) and \( A_{11} \) for the matrix \( A \) defined by the elements \( a_{ij} = 2i j - j \). Here's a step-by-step breakdown:

### Step 1: Define the Matrix \( A \)

The elements of the matrix \( A \) can be calculated using the formula given:

\[ a_{ij} = 2ij - j \]

For a \( 2 \times 2 \) matrix, we have:

- \( a_{11} = 2(1)(1) - 1 = 2 - 1 = 1 \)
- \( a_{12} = 2(1)(2) - 2 = 4 - 2 = 2 \)
- \( a_{21} = 2(2)(1) - 1 = 4 - 1 = 3 \)
- \( a_{22} = 2(2)(2) - 2 = 8 - 2 = 6 \)

Thus, the matrix \( A \) is:

\[ A = \begin{pmatrix} 1 & 2 \\ 3 & 6 \end{pmatrix} \]

### Step 2: Calculate the Minor \( M_{21} \)

The minor \( M_{ij} \) of an element \( a_{ij} \) in a matrix is the determinant of the submatrix formed by deleting the \( i \)-th row and \( j \)-th column.

For \( M_{21} \), we delete the second row and first column:

\[ M_{21} = \text{det} \begin{pmatrix} 2 \end{pmatrix} = 2 \]

### Step 3: Calculate the Cofactor \( A_{11} \)

The cofactor \( A_{ij} \) is given by:

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

For \( A_{11} \):

\[ A_{11} = (-1)^{1+1} M_{11} \]

Now, we need to find \( M_{11} \):

To find \( M_{11} \), we delete the first row and first column:

\[ M_{11} = \text{det} \begin{pmatrix} 6 \end{pmatrix} = 6 \]

Thus,

\[ A_{11} = (-1)^{2} \cdot 6 = 6 \]

### Step 4: Calculate \( M_{21} + A_{11} \)

Now we can find \( M_{21} + A_{11} \):

\[ M_{21} + A_{11} = 2 + 6 = 8 \]

### Final Answer

Thus, the final answer is:

\[ M_{21} + A_{11} = 8 \]