91影视

A non-invertible matrix is defined as a matrix that does not have an inverse, i.e. it does not satisfy the requisite condition to be invertible and is called singular or degenerate matrices. Any non-invertible matrix has a determinant equal to zero.

Given a matrix

$A=\left[\begin{array}{cc}1& -2\\ 2& 1\end{array}\right]$

We start by calculating $A^2$.

$$\begin{gathered} A^2=\left[\begin{array}{cc}1& -2\\ 2& 1\end{array}\right]路\left[\begin{array}{cc}1& -2\\ 2& 1\end{array}\right] \\ =\left[\begin{array}{cc}-3& -4\\ 4& -3\end{array}\right] \end{gathered}$$

Now using exercise 75 we will pick some non-zero vector, we鈥檒l use

$\stackrel{鈫�}{v}=\left[\begin{array}{c}0\\ 1\end{array}\right]$

We will substitute the value of $\stackrel{鈫�}{v}$in $c_0{I}_2\stackrel{鈫�}{v}+c_{1}A\stackrel{鈫�}{v}+c_2{A}^2\stackrel{鈫�}{v}=0$

$c_0{I}_2\stackrel{鈫�}{v}+c_{1}A\stackrel{鈫�}{v}+c_2{A}^2\stackrel{鈫�}{v}=0$

role="math" localid="1664351909495" $c_0.\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right]+c_1.\left[\begin{array}{cc}1& -2\\ 2& 1\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right]+c_2.\left[\begin{array}{cc}-3& -4\\ 4& -3\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right]=0$

$c_0.\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right]+c_1.\left[\begin{array}{cc}1& -2\\ 2& 1\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right]+c_2.\left[\begin{array}{cc}-3& -4\\ 4& -3\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right]=0$

From here

role="math" localid="1664352028948" $$\begin{gathered} c_0路\left[\begin{array}{c}0\\ 1\end{array}\right]+c_1路\left[\begin{array}{c}-2\\ 1\end{array}\right]+c_2路\left[\begin{array}{c}-4\\ -3\end{array}\right]=0 \\ 鈫�\left\{-2c_1-4c_2=0 \\ {c}_0+c_1-3c_2=0\right. \end{gathered}$$

$$\begin{gathered} 鈫�\left\{c_1=-2c_2 \\ {c}_0+c_1-3c_2=0\right. \end{gathered}$$

Now we choose arbitrary $c_2$let鈥檚 say and we鈥檒l get

$c0+c_{1}A+c_2{A}^2=5I_2-2A+A^2=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$