# Math-Magic Tree top row

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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, […] • Top 10 Popular Math Problems in 2016-2017 It's been a year since I started this math blog!! More than 500 problems were posted during a year (July 19th 2016-July 19th 2017). I made a list of the 10 math problems on this blog that have the most views. Can you solve all of them? The level of difficulty among the top […] • If the Augmented Matrix is Row-Equivalent to the Identity Matrix, is the System Consistent? Consider the following system of linear equations: \begin{align*} ax_1+bx_2 &=c\\ dx_1+ex_2 &=f\\ gx_1+hx_2 &=i. \end{align*} (a) Write down the augmented matrix. (b) Suppose that the augmented matrix is row equivalent to the identity matrix. Is the system consistent? […] • Elementary Questions about a Matrix Let $A=\begin{bmatrix} -5 & 0 & 1 & 2 \\ 3 &8 & -3 & 7 \\ 0 & 11 & 13 & 28 \end{bmatrix}.$ (a) What is the size of the matrixA$? (b) What is the third column of$A$? (c) Let$a_{ij}$be the$(i,j)$-entry of$A$. Calculate$a_{23}-a_{31}$. […] • For What Values of$a$, Is the Matrix Nonsingular? Determine the values of a real number$asuch that the matrix $A=\begin{bmatrix} 3 & 0 & a \\ 2 &3 &0 \\ 0 & 18a & a+1 \end{bmatrix}$ is nonsingular. Solution. We apply elementary row operations and obtain: \begin{align*} A=\begin{bmatrix} 3 & 0 & a […] • The Set of Vectors Perpendicular to a Given Vector is a Subspace Fix the row vector\mathbf{b} = \begin{bmatrix} -1 & 3 & -1 \end{bmatrix}$, and let$\R^3$be the vector space of$3 \times 1$column vectors. Define $W = \{ \mathbf{v} \in \R^3 \mid \mathbf{b} \mathbf{v} = 0 \}.$ Prove that$W$is a vector subspace of$\R^3$. […] • If a Symmetric Matrix is in Reduced Row Echelon Form, then Is it Diagonal? Recall that a matrix$A$is symmetric if$A^\trans = A$, where$A^\trans$is the transpose of$A$. Is it true that if$A$is a symmetric matrix and in reduced row echelon form, then$A$is diagonal? If so, prove it. Otherwise, provide a counterexample. Proof. […] • Find All 3 by 3 Reduced Row Echelon Form Matrices of Rank 1 and 2 (a) Find all$3 \times 3$matrices which are in reduced row echelon form and have rank 1. (b) Find all such matrices with rank 2. Solution. (a) Find all$3 \times 3\$ matrices which are in reduced row echelon form and have rank 1. First we look at the rank 1 case. […]