Purdue-Algebra-Exam-eye-catch

• Matrix Operations with Transpose Calculate the following expressions, using the following matrices: $A = \begin{bmatrix} 2 & 3 \\ -5 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 0 & -1 \\ 1 & -1 \end{bmatrix}, \qquad \mathbf{v} = \begin{bmatrix} 2 \\ -4 \end{bmatrix}$ (a) $A B^\trans + \mathbf{v} […] • Find All the Eigenvalues of$A^k$from Eigenvalues of$A$Let$A$be$n\times n$matrix and let$\lambda_1, \lambda_2, \dots, \lambda_n$be all the eigenvalues of$A$. (Some of them may be the same.) For each positive integer$k$, prove that$\lambda_1^k, \lambda_2^k, \dots, \lambda_n^k$are all the eigenvalues of […] • Matrices Satisfying the Relation$HE-EH=2E$Let$H$and$E$be$n \times n$matrices satisfying the relation $HE-EH=2E.$ Let$\lambda$be an eigenvalue of the matrix$H$such that the real part of$\lambda$is the largest among the eigenvalues of$H$. Let$\mathbf{x}$be an eigenvector corresponding to$\lambda$. Then […] • Find a Row-Equivalent Matrix which is in Reduced Row Echelon Form and Determine the Rank For each of the following matrices, find a row-equivalent matrix which is in reduced row echelon form. Then determine the rank of each matrix. (a)$A = \begin{bmatrix} 1 & 3 \\ -2 & 2 \end{bmatrix}$. (b)$B = \begin{bmatrix} 2 & 6 & -2 \\ 3 & -2 & 8 \end{bmatrix}$. (c)$C […]
• Two Subspaces Intersecting Trivially, and the Direct Sum of Vector Spaces. Let $V$ and $W$ be subspaces of $\R^n$ such that $V \cap W =\{\mathbf{0}\}$ and $\dim(V)+\dim(W)=n$. (a) If $\mathbf{v}+\mathbf{w}=\mathbf{0}$, where $\mathbf{v}\in V$ and $\mathbf{w}\in W$, then show that $\mathbf{v}=\mathbf{0}$ and $\mathbf{w}=\mathbf{0}$. (b) If $B_1$ is a […]
• Given the Characteristic Polynomial, Find the Rank of the Matrix Let $A$ be a square matrix and its characteristic polynomial is given by $p(t)=(t-1)^3(t-2)^2(t-3)^4(t-4).$ Find the rank of $A$. (The Ohio State University, Linear Algebra Final Exam Problem)   Solution. Note that the degree of the characteristic polynomial […]
• Possibilities For the Number of Solutions for a Linear System Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer. (a) $\left\{ \begin{array}{c} ax+by=c \\ dx+ey=f, \end{array} \right.$ where $a,b,c, d$ […]