Show that thematrix A=`[{:(,2,3),(,1,2):}]` satisfies the equations `A^(2)-4A+I=0` where I is `2 xx 2` identity matrix and O is `2 xx 2` zero matrix. Using the equations. Find `A^(-1)`.

Explanation

To show that the matrix \( A = \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix} \) satisfies the equation \( A^2 - 4A + I = 0 \) and to find \( A^{-1} \), we will proceed step by step.

### Step 1: Calculate \( A^2 \)
To find \( A^2 \), we need to multiply matrix \( A \) by itself.

\[ A^2 = A \cdot A = \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix} \cdot \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix} \]
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