# cropped-question-logo.jpg

• Compute the Determinant of a Magic Square Let $A= \begin{bmatrix} 8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2 \end{bmatrix} .$ Notice that $A$ contains every integer from $1$ to $9$ and that the sums of each row, column, and diagonal of $A$ are equal. Such a grid is sometimes called a magic […]
• Ring is a Filed if and only if the Zero Ideal is a Maximal Ideal Let $R$ be a commutative ring. Then prove that $R$ is a field if and only if $\{0\}$ is a maximal ideal of $R$.   Proof. $(\implies)$: If $R$ is a field, then $\{0\}$ is a maximal ideal Suppose that $R$ is a field and let $I$ be a non zero ideal: $\{0\} […] • Invertible Idempotent Matrix is the Identity Matrix A square matrix A is called idempotent if A^2=A. Show that a square invertible idempotent matrix is the identity matrix. Proof. Let A be an n \times n invertible idempotent matrix. Since A is invertible, the inverse matrix A^{-1} of A exists and it […] • Ascending Chain of Submodules and Union of its Submodules Let R be a ring with 1. Let M be an R-module. Consider an ascending chain \[N_1 \subset N_2 \subset \cdots$ of submodules of $M$. Prove that the union $\cup_{i=1}^{\infty} N_i$ is a submodule of $M$.   Proof. To simplify the notation, let us […]
• Give the Formula for a Linear Transformation from $\R^3$ to $\R^2$ Let $T: \R^3 \to \R^2$ be a linear transformation such that $T(\mathbf{e}_1)=\begin{bmatrix} 1 \\ 4 \end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix} 2 \\ 5 \end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix} 3 \\ 6 […] • Conjugate of the Centralizer of a Set is the Centralizer of the Conjugate of the Set Let X be a subset of a group G. Let C_G(X) be the centralizer subgroup of X in G. For any g \in G, show that gC_G(X)g^{-1}=C_G(gXg^{-1}). Proof. (\subset) We first show that gC_G(X)g^{-1} \subset C_G(gXg^{-1}). Take any h\in C_G(X). Then for […] • Determine a Condition on a, b so that Vectors are Linearly Dependent Let \[\mathbf{v}_1=\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} 1 \\ a \\ 5 \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 0 \\ 4 \\ b \end{bmatrix}$ be vectors in $\R^3$. Determine a […]
• Two Matrices are Nonsingular if and only if the Product is Nonsingular An $n\times n$ matrix $A$ is called nonsingular if the only vector $\mathbf{x}\in \R^n$ satisfying the equation $A\mathbf{x}=\mathbf{0}$ is $\mathbf{x}=\mathbf{0}$. Using the definition of a nonsingular matrix, prove the following statements. (a) If $A$ and $B$ are \$n\times […]