Find the determinant; operate on square matrices.

The determinant is a scalar value in mathematics that is a function of a square matrix's entries. It allows you to describe some of the matrix's attributes as well as the linear map it represents.

The determinant of square matrices needs to be determined.

Check if the determinant of a matrix is a linear operator.

A linear operator satisfies the condition.

$$\begin{gathered} \mathrm{O}\left(\mathrm{A}+\mathrm{B}\right)=\mathrm{O}\left(\mathrm{A}\right)+\mathrm{O}\left(\mathrm{B}\right) \\ \mathrm{O}\left(\mathrm{kA}\right)=\mathrm{kO}\left(\mathrm{A}\right) \end{gathered}$$

Take for example$2\times 2$matrices of the type.

$\mathrm{M}=\left(\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right)$

Find the determinant.

$\det \left(\mathrm{M}\right)=\mathrm{ad}-\mathrm{bc}$

Take a sum of matrices M and M'.

$\left(\begin{array}{cc}\mathrm{a}+\mathrm{a}\text{'}& \mathrm{b}+\mathrm{b}\text{'}\\ \mathrm{c}+\mathrm{c}\text{'}& \mathrm{d}+\mathrm{d}\text{'}\end{array}\right)$

Find the determinant of M and M'.

localid="1658990437933" $$\begin{gathered} \det \left(\mathrm{M}+\mathrm{M}\text{'}\right)=\left(\mathrm{a}+\mathrm{a}\text{'}\right)\left(\mathrm{d}+\mathrm{d}\text{'}\right)‐\left(\mathrm{b}-\mathrm{b}\text{'}\right)\left(\mathrm{c}+\mathrm{c}\text{'}\right) \\ =\mathrm{ad}‐\mathrm{bc}+\mathrm{a}\text{'}\mathrm{d}\text{'}‐\mathrm{b}\text{'}\mathrm{c}\text{'}+\mathrm{ad}\text{'}+\mathrm{a}\text{'}\mathrm{d}+\mathrm{b}\text{'}\mathrm{c}+\mathrm{bc}\text{'} \\ \ne \mathrm{detM}+\mathrm{detM}\text{'} \end{gathered}$$

The second condition gives the result.

localid="1658990432308" $$\begin{gathered} \det \left(\mathrm{kM}\right)=\left(\mathrm{ka}\right)\left(\mathrm{kd}\right)‐\left(\mathrm{kb}\right)\left(\mathrm{kc}\right) \\ ={\mathrm{k}}^2\mathrm{detM} \end{gathered}$$

Therefore, the determinant is not a linear operator.