# MIT-exam-eye-catch

• Similar Matrices Have the Same Eigenvalues Show that if $A$ and $B$ are similar matrices, then they have the same eigenvalues and their algebraic multiplicities are the same. Proof. We prove that $A$ and $B$ have the same characteristic polynomial. Then the result follows immediately since eigenvalues and algebraic […]
• Image of a Normal Subgroup Under a Surjective Homomorphism is a Normal Subgroup Let $f: H \to G$ be a surjective group homomorphism from a group $H$ to a group $G$. Let $N$ be a normal subgroup of $H$. Show that the image $f(N)$ is normal in $G$.   Proof. To show that $f(N)$ is normal, we show that $gf(N)g^{-1}=f(N)$ for any $g \in […] • Determine Whether There Exists a Nonsingular Matrix Satisfying$A^4=ABA^2+2A^3$Determine whether there exists a nonsingular matrix$A$if $A^4=ABA^2+2A^3,$ where$B$is the following matrix. $B=\begin{bmatrix} -1 & 1 & -1 \\ 0 &-1 &0 \\ 2 & 1 & -4 \end{bmatrix}.$ If such a nonsingular matrix$A$exists, find the inverse […] • Diagonalize a 2 by 2 Symmetric Matrix Diagonalize the$2\times 2$matrix$A=\begin{bmatrix} 2 & -1\\ -1& 2 \end{bmatrix}$by finding a nonsingular matrix$S$and a diagonal matrix$D$such that$S^{-1}AS=D$. Solution. The characteristic polynomial$p(t)$of the matrix$A$[…] • Every Prime Ideal is Maximal if$a^n=a$for any Element$a$in the Commutative Ring Let$R$be a commutative ring with identity$1\neq 0$. Suppose that for each element$a\in R$, there exists an integer$n > 1$depending on$a$. Then prove that every prime ideal is a maximal ideal. Hint. Let$R$be a commutative ring with$1$and$I$be an ideal […] • If Vectors are Linearly Dependent, then What Happens When We Add One More Vectors? Suppose that$\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_r$are linearly dependent$n$-dimensional real vectors. For any vector$\mathbf{v}_{r+1} \in \R^n$, determine whether the vectors$\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_r, \mathbf{v}_{r+1}$are linearly […] • Linear Algebra Midterm 1 at the Ohio State University (3/3) The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017. There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold). The time limit was 55 minutes. This post is Part 3 and contains […] • A Recursive Relationship for a Power of a Matrix Suppose that the$2 \times 2$matrix$A$has eigenvalues$4$and$-2$. For each integer$n \geq 1$, there are real numbers$b_n , c_n$which satisfy the relation $A^{n} = b_n A + c_n I ,$ where$I$is the identity matrix. Find$b_n$and$c_n$for$2 \leq n \leq 5\$, and […]