Question

Find matrices such that$\mathit{d}\mathit{e}\mathit{t}\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]\mathbf{\ne }\left(\det A\right)\left(\det D\right)\mathbf{-}\left(\det B\right)\left(\det C\right)$

Short answer

Therefore,

$\begin{array}{l}A=D=\left[\begin{array}{cc}1& 2\\ 1& 1\end{array}\right]\\ B=c=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]\end{array}$

Step-by-step solution

Definition

A determinant is a unique number associated with asquare matrix.

A determinant is a scalar value that is a function of the entries of a square matrix.

It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

To find the given matrix

For example:

$\begin{array}{l}A=D=\left[\begin{array}{cc}1& 2\\ 1& 1\end{array}\right]\\ B=c=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]\end{array}$

Indeed, we have

$\det A\det D-\det B\det C=\left(-1\right)·\left(-1\right)-0·0$

$\det A\det D-\det B\det C=1$

$\det \left[\begin{array}{cc}A& B\\ C& D\end{array}\right]=\left|\begin{array}{cccc}1& 2& 1& 1\\ 1& 1& 1& 1\\ 1& 1& 1& 2\\ 1& 1& 1& 1\end{array}\right|$

role="math" localid="1660629362263" $\det \left[\begin{array}{cc}A& B\\ C& D\end{array}\right]=0$

Therefore,

$\det \left[\begin{array}{cc}A& B\\ C& D\end{array}\right]\ne \left(\det A\right)\left(\det D\right)-\left(\det B\right)\left(\det C\right)$

Therefore,

$\begin{array}{l}A=D=\left[\begin{array}{cc}1& 2\\ 1& 1\end{array}\right]\\ B=c=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]\end{array}$