# Math-Magic Tree filled

• The Trick of a Mathematical Game. The One’s Digit of the Sum of Two Numbers. Decipher the trick of the following mathematical magic.   The Rule of the Game Here is the game. Pick six natural numbers ($1, 2, 3, \dots$) and place them in the yellow discs of the picture below. For example, let's say I have chosen the numbers $7, 5, 3, 2, […] • The Union of Two Subspaces is Not a Subspace in a Vector Space Let$U$and$V$be subspaces of the vector space$\R^n$. If neither$U$nor$V$is a subset of the other, then prove that the union$U \cup V$is not a subspace of$\R^n$. Proof. Since$U$is not contained in$V$, there exists a vector$\mathbf{u}\in U$but […] • Inverse Matrix Contains Only Integers if and only if the Determinant is$\pm 1$Let$A$be an$n\times n$nonsingular matrix with integer entries. Prove that the inverse matrix$A^{-1}$contains only integer entries if and only if$\det(A)=\pm 1$. Hint. If$B$is a square matrix whose entries are integers, then the […] • Galois Group of the Polynomial$x^2-2$Let$\Q$be the field of rational numbers. (a) Is the polynomial$f(x)=x^2-2$separable over$\Q$? (b) Find the Galois group of$f(x)$over$\Q$. Solution. (a) The polynomial$f(x)=x^2-2$is separable over$\Q$The roots of the polynomial$f(x)$are$\pm […]
• Example of a Nilpotent Matrix $A$ such that $A^2\neq O$ but $A^3=O$. Find a nonzero $3\times 3$ matrix $A$ such that $A^2\neq O$ and $A^3=O$, where $O$ is the $3\times 3$ zero matrix. (Such a matrix is an example of a nilpotent matrix. See the comment after the solution.)   Solution. For example, let $A$ be the following $3\times […] • Questions About the Trace of a Matrix Let$A=(a_{i j})$and$B=(b_{i j})$be$n\times n$real matrices for some$n \in \N$. Then answer the following questions about the trace of a matrix. (a) Express$\tr(AB^{\trans})$in terms of the entries of the matrices$A$and$B$. Here$B^{\trans}$is the transpose matrix of […] • Determine linear transformation using matrix representation Let$T$be the linear transformation from the$3$-dimensional vector space$\R^3$to$\R^3itself satisfying the following relations. \begin{align*} T\left(\, \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \,\right) =\begin{bmatrix} 1 \\ 0 \\ 1 […] • The Formula for the Inverse Matrix ofI+A$for a$2\times 2$Singular Matrix$A$Let$A$be a singular$2\times 2$matrix such that$\tr(A)\neq -1$and let$I$be the$2\times 2$identity matrix. Then prove that the inverse matrix of the matrix$I+A\$ is given by the following formula: $(I+A)^{-1}=I-\frac{1}{1+\tr(A)}A.$ Using the formula, calculate […]