Question

Let \( P = (a_{ij}) \) be a \( 10 \times 10 \) matrix with \[ a_{ij} = \begin{cases} -\frac{1}{10} & \text{if } i \neq j, \\ \frac{9}{10} & \text{if } i = j. \end{cases} \] Then the rank of \( P \) equals:

• A. 10
• B. 9
• C. 1
• D. 8

Hint:

For matrices with a dominant diagonal and constant off-diagonal entries, the rank is generally the number of linearly independent rows. In this case, it is 9.

Answer 1

The Correct Option is B

Step 1: Matrix Structure
The matrix \( P \) is a \( 10 \times 10 \) matrix, where the diagonal entries \( a_{ii} = \frac{9}{10} \) and the off-diagonal entries \( a_{ij} = -\frac{1}{10} \) for \( i \neq j \). We can write the matrix as: \[ P = \frac{1}{10} \begin{pmatrix} 9 & -1 & -1 & \cdots & -1 \\ -1 & 9 & -1 & \cdots & -1 \\ -1 & -1 & 9 & \cdots & -1 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ -1 & -1 & -1 & \cdots & 9 \end{pmatrix}. \] This matrix is a special form where the diagonal entries dominate, and it is highly structured.

Step 2: Observation on Linear Dependence
The rows of \( P \) are linearly dependent because the off-diagonal elements are constant and the diagonal elements are all equal. The rows of the matrix can be seen as a combination of a constant vector and a constant diagonal term. This leads to the fact that the rank of \( P \) will be less than 10.

Step 3: Rank of Matrix
To determine the rank, observe that there are essentially 9 independent rows in this matrix, with the rows being highly similar to each other. The rank of such a matrix is typically \( 9 \) because of the structure.

Thus, the rank of \( P \) is \( \boxed{9} \).