Question

Matrix A as a product of two other matrices is given by \[ A=\begin{bmatrix} 3 \\ 2 \end{bmatrix} \begin{bmatrix} 1\;\;4 \end{bmatrix}. \] The value of det(A) is \(\underline{\hspace{3cm}}\) (round off to nearest integer).

Hint:

A rank-1 matrix formed as $uv^T$ always has determinant zero because its rows (or columns) are linearly dependent.

Answer 1

Correct Answer: 0

Matrix $A$ is formed by multiplying a $2\times 1$ matrix with a $1\times 2$ matrix: \[ A= \begin{bmatrix} 3 \\ 2 \end{bmatrix} \begin{bmatrix} 1 & 4 \end{bmatrix} = \begin{bmatrix} 3\cdot 1 & 3\cdot 4 \\ 2\cdot 1 & 2\cdot 4 \end{bmatrix} = \begin{bmatrix} 3 & 12 \\ 2 & 8 \end{bmatrix}. \] The determinant of a $2\times 2$ matrix \[ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \] is \[ \det(A)=ad-bc. \] Thus, \[ \det(A)=3\times 8 - 12\times 2 = 24 - 24 = 0. \] Rounded to the nearest integer, the determinant is: \[ 0. \]