In this chapter, we mainly focus on square matrices.[br][br]Let [math]A[/math] be an n x n matrix. We want to find a special nonzero vector in [math]\mathbb{R}^n[/math] such that its transformation by [math]A[/math] is a scaling of this vector. This nonzero vector is called an [b]eigenvector[/b] and the scaling factor is called an [b]eigenvalue[/b] (note: it can be zero). More rigorously, we have the following definition:[br][br][u]Definition[/u]: An [b]eigenvector[/b] of an n x n matrix [math]A[/math] is a [u]nonzero[/u] vector [math]x[/math] such that [math]Ax=\lambda x[/math] for some real number [math]\lambda[/math], which is called an [b]eigenvalue[/b] of [math]A[/math]. [math]x[/math] is said to be an eigenvector corresponding to [math]\lambda[/math].[br][br]From the geometric point of view, it means that the line containing vector [math]x[/math] i.e.[math]\text{Span}\{x\}[/math] remains unchanged under the linear transformation corresponding to [math]A[/math]. And any vector in [math]\text{Span}\{x\}[/math] will be scaled by factor [math]\lambda[/math] under the transformation. [br][br]In the applet below, you first define the 2 x 2 matrix [math]A[/math] by setting [math]T(\hat{\mathbf{i}})[/math] and [math]T(\hat{\mathbf{j}})[/math], you need to move the vector such that it lies on the orange dotted line containing [math]Av[/math].[br][br]Complete the following tasks:[br][list=1][*]Find a matrix [math]A[/math] that has two eigenvalues.[/*][*]Find a matrix [math]A[/math] that has only one eigenvalue and all the eigenvectors lie on a line.[/*][*]Find a matrix [math]A[/math] that has only one eigenvalue and all nonzero vectors are eigenvectors.[/*][*]Find a matrix [math]A[/math] that has no eigenvalue. [br][/*][/list][br][br]