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\[\color {purple} {Example\ 1:}\ \color {red}{Find\ the\ determinant\ of}\ A =\begin{vmatrix}
2 & -5 \\
1 & 3 \\
\end{vmatrix}\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ A = \begin{vmatrix}
2 & -5 \\
1 & 3 \\
\end{vmatrix}\ \hspace{15cm}\]
\[\begin{vmatrix}
A \\
\end{vmatrix}\ = 2(3) – (-5)(1) \hspace{13cm}\]
\[= 6 + 5 \hspace{12cm}\]
\[\Delta= 11 \hspace{12cm}\]
\[\color {purple} {Example\ 2\ :}\ \color {red}{Solve}\ \begin{vmatrix}
x & 2 \\
2 & x \\
\end{vmatrix}\ = 0\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ \begin{vmatrix}
x & 2 \\
2 & x \\
\end{vmatrix}\ = 0 \hspace{15cm}\]
\[x(x) – 2 (2) = 0\ \hspace{13cm}\]
\[x^2 – 4 = 0\ \hspace{13cm}\]
\[x^2 = 4\ \hspace{13cm}\]
\[x = \pm\sqrt{4}\ \hspace{13cm}\]
\[x = \pm 2\ \hspace{13cm}\]
\[\color {purple} {Example\ 3\ :}\ \color {red}{Evaluate}\ \begin{vmatrix}
sin\ \theta & -\ cos\ \theta \\
cos\ \theta & sin\ \theta \\
\end{vmatrix}\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ \begin{vmatrix}
sin\ \theta & -\ cos\ \theta \\
cos\ \theta & sin\ \theta \\
\end{vmatrix}\ \hspace{15cm}\]
\[=\ sin\ \theta(sin\ \theta)\ -\ cos\ \theta (-\ cos\ \theta)\ \hspace{13cm}\]
\[=\ sin^2\ \theta\ +\ cos^2\ \theta \ \hspace{13cm}\]
\[=\ 1\ \hspace{13cm}\]
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