# Math-Magic Tree Trick

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• 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 $7, 5, 3, 2, […] • Conditions on Coefficients that a Matrix is Nonsingular (a) Let$A=(a_{ij})$be an$n\times n$matrix. Suppose that the entries of the matrix$A$satisfy the following relation. $|a_{ii}|>|a_{i1}|+\cdots +|a_{i\,i-1}|+|a_{i \, i+1}|+\cdots +|a_{in}|$ for all$1 \leq i \leq n$. Show that the matrix$A$is nonsingular. (b) Let […] • Use the Cayley-Hamilton Theorem to Compute the Power$A^{100}$Let$A$be a$3\times 3$real orthogonal matrix with$\det(A)=1$. (a) If$\frac{-1+\sqrt{3}i}{2}$is one of the eigenvalues of$A$, then find the all the eigenvalues of$A$. (b) Let $A^{100}=aA^2+bA+cI,$ where$I$is the$3\times 3$identity matrix. Using the […] • Compute and Simplify the Matrix Expression Including Transpose and Inverse Matrices Let$A, B, C$be the following$3\times 3$matrices. $A=\begin{bmatrix} 1 & 2 & 3 \\ 4 &5 &6 \\ 7 & 8 & 9 \end{bmatrix}, B=\begin{bmatrix} 1 & 0 & 1 \\ 0 &3 &0 \\ 1 & 0 & 5 \end{bmatrix}, C=\begin{bmatrix} -1 & 0\ & 1 \\ 0 &5 &6 \\ 3 & 0 & […] • Cosine and Sine Functions are Linearly Independent Let C[-\pi, \pi] be the vector space of all continuous functions defined on the interval [-\pi, \pi]. Show that the subset \{\cos(x), \sin(x)\} in C[-\pi, \pi] is linearly independent. Proof. Note that the zero vector in the vector space C[-\pi, \pi] is […] • Find the Matrix Representation of T(f)(x) = f(x^2) if it is a Linear Transformation For an integer n > 0, let \mathrm{P}_n denote the vector space of polynomials with real coefficients of degree 2 or less. Define the map T : \mathrm{P}_2 \rightarrow \mathrm{P}_4 by \[ T(f)(x) = f(x^2).$ Determine if$T$is a linear transformation. If it is, find […] • Non-Prime Ideal of Continuous Functions Let$R$be the ring of all continuous functions on the interval$[0,1]$. Let$I$be the set of functions$f(x)$in$R$such that$f(1/2)=f(1/3)=0$. Show that the set$I$is an ideal of$R$but is not a prime ideal. Proof. We first show that$I$is an ideal of […] • The Range and Null Space of the Zero Transformation of Vector Spaces Let$U$and$V$be vector spaces over a scalar field$\F$. Define the map$T:U\to V$by$T(\mathbf{u})=\mathbf{0}_V$for each vector$\mathbf{u}\in U$. (a) Prove that$T:U\to V$is a linear transformation. (Hence,$T\$ is called the zero transformation.) (b) Determine […]