Chapter 12.3, Problem 61PPSE

Solution

To prove: That the product of the determinants of $M$ and $N$ equals the determinant of the matrix product $MN$ .

Given information: Matrices are $M=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ and $N=\left[\begin{array}{cc}e& f\\ g& h\end{array}\right]$ .

Formula used: Matrix multiplication is,

  $\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]\left[\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right]=\left[\begin{array}{cc}{a}_{11}{b}_{11}+a_{12}{b}_{21}& a_{11}{b}_{12}+a_{12}{b}_{22}\\ a_{21}{b}_{11}+a_{22}{b}_{21}& a_{21}{b}_{12}+a_{22}{b}_{22}\end{array}\right]$ .

Determinant of the matrix $\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]$ is,

  $\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|=a_{11}{a}_{22}-a_{12}{a}_{21}$ .

Calculation:

Consider the matrices as $M=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ and $N=\left[\begin{array}{cc}e& f\\ g& h\end{array}\right]$ .

Then,

  $\begin{array}{c}MN=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}e& f\\ g& h\end{array}\right]\\ =\left[\begin{array}{cc}ae+bg& af+bh\\ ce+dg& cf+dh\end{array}\right]\end{array}$ .

Now, determinants of $M$ and $N$ are,

  $\begin{array}{c}\left|M\right|=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|\\ =ad-bc\end{array}$ .

And,

  $\begin{array}{c}\left|N\right|=\left|\begin{array}{cc}e& f\\ g& h\end{array}\right|\\ =eh-gf\end{array}$ .

Then the product of the determinants of $M$ and $N$ is,

  $\begin{array}{c}\left|M\right|\left|N\right|=\left(ad-bc\right)\left(eh-gf\right)\\ =adeh-adgf-bceh+bcgf\end{array}$ .

The determinant of $MN$ is,

  $\begin{array}{c}\left|MN\right|=\left|\begin{array}{cc}ae+bg& af+bh\\ ce+dg& cf+dh\end{array}\right|\\ =\left(ae+bg\right)\left(cf+dh\right)-\left(af+bh\right)\left(ce+dg\right)\\ =aecf+aedh+bgcf+bgdh-afce-afdg-bhce-bhdg\\ =aedh+bgcf-afdg-bhce\end{array}$ .

Therefore, the product of the determinants of $M$ and $N$ equals the determinant of the matrix product $MN$ .