Question

Consider the row vectors \( v = (1, 0) \) and \( w = (2, 0) \). The rank of the matrix \[ M = 2v^T v + 3w^T w \] where the superscript \( T \) denotes the transpose, is

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

The rank of a matrix is determined by the number of linearly independent rows or columns.

Answer 1

The Correct Option is A

We are given the row vectors \( v = (1, 0) \) and \( w = (2, 0) \), and we need to find the rank of the matrix \( M = 2v^T v + 3w^T w \). Step 1: Calculate the matrix products. \\ We first compute the transposes of \( v \) and \( w \): \[ v^T = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, w^T = \begin{pmatrix} 2 \\ 0 \end{pmatrix} \] Now compute the matrix products: \[ 2v^T v = 2 \begin{pmatrix} 1 \\ 0 \end{pmatrix} \begin{pmatrix} 1 & 0 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & 0 \end{pmatrix} \] \[ 3w^T w = 3 \begin{pmatrix} 2 \\ 0 \end{pmatrix} \begin{pmatrix} 2 & 0 \end{pmatrix} = \begin{pmatrix} 12 & 0 \\ 0 & 0 \end{pmatrix} \] Step 2: Add the matrices. \[ M = \begin{pmatrix} 2 & 0 \\ 0 & 0 \end{pmatrix} + \begin{pmatrix} 12 & 0 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 14 & 0 \\ 0 & 0 \end{pmatrix} \] Step 3: Find the rank of the matrix. The rank of the matrix is the number of non-zero rows (or columns). Since there is only one non-zero row in \( M \), the rank is 1. Therefore, the correct answer is option (A). Final Answer: 1 \\