Question

The function $\mathbf{q}\left(\overrightarrow{x}\right)\mathbf{=}\stackrel{\mathbf{\to }\mathbf{T}}{\mathbf{x}}\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\overrightarrow{\mathbf{x}}$is a quadratic form.

Short answer

The given statement is TRUE.

Step-by-step solution

Given information:

The function $\mathrm{q}\left(\overrightarrow{\mathrm{x}}\right)=\stackrel{\to \mathrm{T}}{\mathrm{x}}\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\overrightarrow{\mathrm{x}}$is a quadratic form.

Check whether the given statement is TRUE or FALSE

It is known that,a matrix A such that $\mathbf{q}\left(\overrightarrow{x}\right)\mathbf{=}\stackrel{\mathbf{\to }\mathbf{T}}{\mathbf{x}}\mathbf{A}\overrightarrow{\mathbf{x}}$denotes a quadratic form.

The given matrix is$2\times 2$matrix. It has two variables.

Also, $\mathrm{x}=\left[\begin{array}{c}{\mathrm{x}}_1\\ {\mathrm{x}}_2\end{array}\right]\mathrm{and}{\mathrm{x}}^{\mathrm{T}}=\left[{\mathrm{x}}_1{\mathrm{x}}_2\right]$

Substitute these values in the given equation and simplify:

$$\begin{gathered} \mathrm{q}\left(\mathrm{x}\right)={\mathrm{x}}^{\mathrm{T}}\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\mathrm{x} \\ =\left[{\mathrm{x}}_1{\mathrm{x}}_2\right]\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\left[\begin{array}{c}{\mathrm{x}}_1\\ {\mathrm{x}}_2\end{array}\right] \\ =\left[{\mathrm{x}}_1{\mathrm{x}}_2\right]\left[\begin{array}{c}{\mathrm{x}}_1+2{\mathrm{x}}_2\\ 3{\mathrm{x}}_1+4{\mathrm{x}}_2\end{array}\right] \\ ={\mathrm{x}}_1^2+2{\mathrm{x}}_1{\mathrm{x}}_2+3{\mathrm{x}}_1{\mathrm{x}}_2+4{\mathrm{x}}_2^2 \\ ={\mathrm{x}}_1^2+5{\mathrm{x}}_1{\mathrm{x}}_2+4{\mathrm{x}}_2^2 \end{gathered}$$

Here, $\mathrm{q}\left(\mathrm{x}\right)$is in a quadratic form.

Final Answer

The given statement is TRUE.