Question

Let \( M = \sum_{i=1}^{4} X_i X_i^T \), where \[ X_1^T = [1 \ -1 \ 1 \ 0], X_2^T = [1 \ 1 \ 0 \ 1], X_3^T = [1 \ 3 \ 1 \ 0] \, \text{and} \, X_4^T = [1 \ 1 \ 1 \ 0]. \] Then the rank of \( M \) equals ............... 
 

Hint:

The rank of a matrix formed by summing outer products of vectors is the number of linearly independent vectors in the set.

Answer 1

Correct Answer: 3

Step 1: Understand the matrix \( M \).
The matrix \( M \) is the sum of outer products of the vectors \( X_i \). Each vector \( X_i \) is a row vector, and the outer product \( X_i X_i^T \) will be a matrix of rank 1.

Step 2: Construct the matrix \( M \).
We calculate the outer product for each \( X_i \): - \( X_1 X_1^T = \begin{bmatrix} 1 \\ -1 \\ 1 \\ 0 \end{bmatrix} \begin{bmatrix} 1 & -1 & 1 & 0 \end{bmatrix} = \begin{bmatrix} 1 & -1 & 1 & 0 \\ -1 & 1 & -1 & 0 \\ 1 & -1 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \) - Similarly for \( X_2, X_3, X_4 \).

Step 3: Determine the rank of \( M \).
After summing the matrices, we observe that the resulting matrix has 3 linearly independent rows, so the rank of \( M \) is 3.

Step 4: Conclusion.
Thus, the rank of \( M \) is \( \boxed{3} \).