Question

If the determinant of a$\mathbf{2}\mathbf{\times }\mathbf{2}$ matrix is 4, then the inequality $\left|\left|A\overrightarrow{v}\right|\right|\mathbf{\le }\mathbf{4}\left|\left|\overrightarrow{v}\right|\right|$ must hold for all vectors $\overrightarrow{\mathbf{v}}$ in.

Short answer

Therefore, the given statement is true.

Step-by-step solution

 Step 1: Matrix Definition

Matrix is aset of numbers arranged in rows and columns so as to form a rectangulararray.

The numbers are called the elements, or entries, of the matrix.

If there are m rows and n columns, the matrix is said to be an “m by n ” matrix, written “ $m\times n$.”

To check whether the given condition is true or false

Let,

$A=\left[\begin{array}{cc}4& 4\\ 0& 1\end{array}\right]$.

Clearly,

det A = 4 , however, for

$V=\left[\begin{array}{c}1\\ 1\end{array}\right]$

We have,

$$\begin{gathered} \left|\left|Av\right|\right|=\left|\left|\left[\left[\begin{array}{c}8\\ 1\end{array}\right]\right]\right|\right| \\ \left|\left|Av\right|\right|=\sqrt{65}>\sqrt{32} \\ \left|\left|Av\right|\right|=4\left|\left|v\right|\right| \end{gathered}$$

Therefore, the given statement is true.