For `A 2 times 2` matrix, whose elements are given by: `a_(ij)=1/2abs(-3i+j)`. Find `a_(11)+a_(22)`

Answer

To solve the problem, we need to find the elements \( a_{11} \) and \( a_{22} \) of the matrix defined by the formula \( a_{ij} = \frac{1}{2} | -3i + j | \). We will then compute the sum \( a_{11} + a_{22} \).

### Step 1: Calculate \( a_{11} \)

Using the formula for \( a_{ij} \):

\[ a_{11} = \frac{1}{2} | -3(1) + 1 | \]

Calculating the expression inside the modulus:

\[ -3(1) + 1 = -3 + 1 = -2 \]

Now, taking the absolute value:

\[ | -2 | = 2 \]

Now substituting back into the formula:

\[ a_{11} = \frac{1}{2} \times 2 = 1 \]

### Step 2: Calculate \( a_{22} \)

Now we calculate \( a_{22} \):

\[ a_{22} = \frac{1}{2} | -3(2) + 2 | \]

Calculating the expression inside the modulus:

\[ -3(2) + 2 = -6 + 2 = -4 \]

Now, taking the absolute value:

\[ | -4 | = 4 \]

Now substituting back into the formula:

\[ a_{22} = \frac{1}{2} \times 4 = 2 \]

### Step 3: Calculate \( a_{11} + a_{22} \)

Now we find the sum:

\[ a_{11} + a_{22} = 1 + 2 = 3 \]

Thus, the final answer is:

\[ \boxed{3} \]

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