Construct a `2xx2` matrix `A=[a_(ij)]` whose elements are given by :
`a_(ij)=1/3|(2i-3j)|`

Answer

To construct the \(2 \times 2\) matrix \(A = [a_{ij}]\) whose elements are given by the formula:

\[ a_{ij} = \frac{1}{3} |2i - 3j| \]

we will calculate each element of the matrix step by step.

### Step 1: Identify the elements of the matrix

The matrix \(A\) has the following structure:

\[ A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \]

where \(i\) and \(j\) take values from 1 to 2.

### Step 2: Calculate \(a_{11}\)

For \(a_{11}\) (where \(i = 1\) and \(j = 1\)):

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

### Step 3: Calculate \(a_{12}\)

For \(a_{12}\) (where \(i = 1\) and \(j = 2\)):

\[ a_{12} = \frac{1}{3} |2(1) - 3(2)| = \frac{1}{3} |2 - 6| = \frac{1}{3} |-4| = \frac{1}{3} \cdot 4 = \frac{4}{3} \]

### Step 4: Calculate \(a_{21}\)

For \(a_{21}\) (where \(i = 2\) and \(j = 1\)):

\[ a_{21} = \frac{1}{3} |2(2) - 3(1)| = \frac{1}{3} |4 - 3| = \frac{1}{3} |1| = \frac{1}{3} \cdot 1 = \frac{1}{3} \]

### Step 5: Calculate \(a_{22}\)

For \(a_{22}\) (where \(i = 2\) and \(j = 2\)):

\[ a_{22} = \frac{1}{3} |2(2) - 3(2)| = \frac{1}{3} |4 - 6| = \frac{1}{3} |-2| = \frac{1}{3} \cdot 2 = \frac{2}{3} \]

### Step 6: Construct the matrix

Now we can construct the matrix \(A\) using the calculated values:

\[ A = \begin{bmatrix} \frac{1}{3} & \frac{4}{3} \\ \frac{1}{3} & \frac{2}{3} \end{bmatrix} \]

### Final Answer

The \(2 \times 2\) matrix \(A\) is:

\[ A = \begin{bmatrix} \frac{1}{3} & \frac{4}{3} \\ \frac{1}{3} & \frac{2}{3} \end{bmatrix} \]