Question

A, B, C and D are vectors of length 4. The rank of the matrix 
A = \(\begin{bmatrix} a_1 & a_2 & a_3 & a_4 \end{bmatrix}\), 
B = \(\begin{bmatrix} b_1 & b_2 & b_3 & b_4 \end{bmatrix}\), 
C = \(\begin{bmatrix} c_1 & c_2 & c_3 & c_4 \end{bmatrix}\), 
D = \(\begin{bmatrix} d_1 & d_2 & d_3 & d_4 \end{bmatrix}\) 
It is known that B is not a scalar multiple of A. Also, C is linearly independent of A and B. Further, \( D = 3A + 2B + C \) 

The rank of the matrix \( \begin{bmatrix} a_1 & a_2 & a_3 & a_4 \\ b_1 & b_2 & b_3 & b_4 \\ c_1 & c_2 & c_3 & c_4 \\ d_1 & d_2 & d_3 & d_4 \end{bmatrix} \) is \(\underline{\hspace{1cm}}\)

Hint:

The rank of a matrix formed by vectors is the number of linearly independent vectors in the matrix. In this case, the vectors \( A \), \( B \), and \( C \) are linearly independent, so the rank of the matrix is 3.

Answer 1

Correct Answer: 3

We are given that \( B \) is not a scalar multiple of \( A \), meaning \( A \) and \( B \) are linearly independent. Also, \( C \) is linearly independent of \( A \) and \( B \), meaning that \( A \), \( B \), and \( C \) are all linearly independent vectors. Now, since \( D = 3A + 2B + C \), the vector \( D \) is a linear combination of \( A \), \( B \), and \( C \). Thus, the matrix formed by these vectors has at most 3 linearly independent columns, as the rank of a matrix is the number of linearly independent columns. Therefore, the rank of the matrix is: \[ \boxed{3} \]