Question

If \( A = [a_{ij}]\) where \( 1 \leq i, j \leq n \) with \( n \geq 2 \) and \( a_{ij} = i + j \) is a matrix, then the rank of \( A \) is:

• A. 0
• B. 1
• C. 2
• D. 4

Hint:

Linear dependence in matrix rows or columns can be inferred if each element in rows or columns can be expressed as a linear combination of others.

Answer 1

The Correct Option is C

Step 1: Matrix \( A \) has elements defined by \( a_{ij} = i + j \). This means that each element in row \( i \) and column \( j \) is the sum of \( i \) and \( j \). 
For example, for the 2x2 matrix, the entries will be: \[ A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} = \begin{bmatrix} 1+1 & 1+2 \\ 2+1 & 2+2 \end{bmatrix} = \begin{bmatrix} 2 & 3 \\ 3 & 4 \end{bmatrix} \] We can observe that the rows are linearly dependent because each row is just a linear combination of the other. In general, this holds for any size of the matrix where the sum \( a_{ij} = i + j \). 
Step 2: To compute the rank, we need to reduce the matrix to row echelon form. Starting with the general form, let’s consider a 3x3 matrix for illustration: \[ A = \begin{bmatrix} 2 & 3 & 4 \\ 3 & 4 & 5 \\ 4 & 5 & 6 \end{bmatrix} \] Using elementary row operations, we can reduce this matrix to row echelon form. After performing row operations, we see that there are only two non-zero rows, indicating that the rank of the matrix is 2. Thus, the rank of \( A \) is 2.