Let A be a $$2 \times2$$​ real matrix with detA=1 and trace A=3.

What is the value of trace of $$A^2?$$​​

Correct option is D

Let A be a $$2\times2$$ matrix with detA=1 and traceA=3.

the characteristic equation is given by:

$$x^2-(traceA)x+detA=0\\[10pt]\textbf{substituting the values we get}:\\[10pt]x^2-3x+1=0\\[10pt]\implies x=\frac{-(-3)\pm \sqrt{9-4}}{2}\\[10pt]\text{Eigen values of A are:} \frac{3+\sqrt{5}}{2},\frac{3-\sqrt{5}}{2}\\[10pt]\text{Eigen values of } A^2=\big( \frac{3+\sqrt{5}}{2}\big)^2,\big( \frac{3-\sqrt{5}}{2}\big)^2\\[10pt]\implies \frac{7+3\sqrt{5}}{2} and \frac{7-3\sqrt{5}}{2}\\[10pt]\textbf{Hence trace wil be 7 (Option D)}$$​