Question

If \( [a_{ij}] = 2i - j \), then determine a matrix A of order \( 2 \times 3 \).

Hint:

When constructing a matrix from a formula, be systematic. Go row by row or column by column, carefully substituting the correct values of \( i \) (row index) and \( j \) (column index) into the given formula.

Answer 1

Step 1: Understanding the Concept:
We are asked to construct a matrix A of order \( 2 \times 3 \). This means the matrix will have 2 rows and 3 columns. The value of each element \( a_{ij} \) in the matrix is given by the formula \( 2i - j \), where \( i \) is the row number and \( j \) is the column number.
Step 2: Key Formula or Approach:
A general \( 2 \times 3 \) matrix A can be written as:
\[ A = \begin{bmatrix} a_{11} & a_{12} & a_{13}
a_{21} & a_{22} & a_{23} \end{bmatrix} \] We will calculate each element \( a_{ij} \) using the given formula \( a_{ij} = 2i - j \).
Step 3: Detailed Explanation or Calculation:
We calculate each of the six elements of the matrix:
For the first row (i=1):
- \( a_{11} = 2(1) - 1 = 2 - 1 = 1 \)
- \( a_{12} = 2(1) - 2 = 2 - 2 = 0 \)
- \( a_{13} = 2(1) - 3 = 2 - 3 = -1 \)
For the second row (i=2):
- \( a_{21} = 2(2) - 1 = 4 - 1 = 3 \)
- \( a_{22} = 2(2) - 2 = 4 - 2 = 2 \)
- \( a_{23} = 2(2) - 3 = 4 - 3 = 1 \)
Now, we assemble these elements into the matrix A.
Step 4: Final Answer:
The matrix A of order \( 2 \times 3 \) is:
\[ A = \begin{bmatrix} 1 & 0 & -1
3 & 2 & 1 \end{bmatrix} \]