Construct a `3xx2` matrix whose elements are given by `a_(ij)=2i-j`

Answer

To construct a \(3 \times 2\) matrix whose elements are given by \(a_{ij} = 2i - j\), we will follow these steps:

### Step 1: Define the Matrix Structure
A \(3 \times 2\) matrix has 3 rows and 2 columns. We will denote the matrix as \(A\) and its elements as \(a_{ij}\), where \(i\) represents the row number and \(j\) represents the column number.

### Step 2: Identify the Elements
Using the formula \(a_{ij} = 2i - j\), we will calculate the elements for each position in the matrix.

### Step 3: Calculate Each Element
1. **For \(i = 1\)**:
- \(a_{11} = 2(1) - 1 = 2 - 1 = 1\)
- \(a_{12} = 2(1) - 2 = 2 - 2 = 0\)

2. **For \(i = 2\)**:
- \(a_{21} = 2(2) - 1 = 4 - 1 = 3\)
- \(a_{22} = 2(2) - 2 = 4 - 2 = 2\)

3. **For \(i = 3\)**:
- \(a_{31} = 2(3) - 1 = 6 - 1 = 5\)
- \(a_{32} = 2(3) - 2 = 6 - 2 = 4\)

### Step 4: Construct the Matrix
Now that we have calculated all the elements, we can construct the matrix \(A\):

\[ A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 3 & 2 \\ 5 & 4 \end{bmatrix} \]

### Final Result
The constructed \(3 \times 2\) matrix is:

\[ A = \begin{bmatrix} 1 & 0 \\ 3 & 2 \\ 5 & 4 \end{bmatrix} \]

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