91影视

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 鈥� m by n 鈥� matrix, written 鈥� $m脳n$.鈥�

Given matrix,

$M_4=\left[\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 1& 0& 0& 0\end{array}\right]$

To solve,

$$\begin{gathered} det\left(M_2\right)=\left|\begin{array}{cc}0& 1\\ 1& 0\end{array}\right| \\ det\left(M_2\right)=-1 \\ det\left(M_3\right)=\left|\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right| \\ det\left(M_3\right)=1.\left|\begin{array}{cc}0& 1\\ 1& 0\end{array}\right| \\ det\left(M_3\right)=1.(-1) \\ det\left(M_3\right)=-1 \end{gathered}$$

$$\begin{gathered} \det \left(M_4\right)=\left|\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 1& 0& 0& 0\end{array}\right| \\ \det \left(M_4\right)=\left|\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right| \\ \det \left(M_4\right)=-1.(-1) \\ \det \left(M_4\right)=1 \end{gathered}$$

$$\begin{gathered} \det \left(M_{\mathit{5}}\right)=\left|\begin{array}{ccccc}0& 0& 0& 0& 1\\ 0& 0& 0& 1& 0\\ 0& 0& 1& 0& 0\\ 0& 1& 0& 0& 0\\ 1& 0& 0& 0& 0\end{array}\right| \\ \det \left(M_{\mathit{5}}\right)=1.\left|\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 1& 0& 0& 0\end{array}\right| \\ \det \left(M_{\mathit{5}}\right)1.1 \\ \det \left(M_{\mathit{5}}\right)=1 \\ \det \left(M_{\mathit{6}}\right)=\left|\begin{array}{cccccc}0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0\end{array}\right| \end{gathered}$$

$$\begin{gathered} \det \left(M_{\mathit{6}}\right)=-1\left|\begin{array}{ccccc}0& 0& 0& 0& 1\\ 0& 0& 0& 1& 0\\ 0& 0& 1& 0& 0\\ 0& 1& 0& 0& 0\\ 1& 0& 0& 0& 0\end{array}\right| \\ \det \left(M_{\mathit{6}}\right)=-1 \end{gathered}$$

role="math" localid="1660111904948" $$\begin{gathered} \det \left(M_{\mathit{7}}\right)=\left|\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0\end{array}\right| \\ \det \left({\mathrm{M}}_{\mathit{7}}\right)=1.\left|\begin{array}{cccccc}0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0\end{array}\right| \\ \det \left({\mathrm{M}}_{\mathit{7}}\right)=1.\left(-1\right) \\ \det \left({\mathrm{M}}_{\mathit{7}}\right)=-1 \end{gathered}$$

If is an even number, then we need $\frac{n}{2}$row interchanges to obtain the identity matrix $I_n$. Then, the determinant is ${\left(-1\right)}^{\frac{n}{2}}$.

If, on the other hand, n is odd, then $I_n$will be obtained after $\frac{n-1}{2}$row interchanges, so then the determinant will be ${\left(-1\right)}^{\frac{n-1}{2}}$.