More from my site

• Rank of the Product of Matrices $AB$ is Less than or Equal to the Rank of $A$ Let $A$ be an $m \times n$ matrix and $B$ be an $n \times l$ matrix. Then prove the followings. (a) $\rk(AB) \leq \rk(A)$. (b) If the matrix $B$ is nonsingular, then $\rk(AB)=\rk(A)$.   Hint. The rank of an $m \times n$ matrix $M$ is the dimension of the range […]
• Find a Nonsingular Matrix Satisfying Some Relation Determine whether there exists a nonsingular matrix $A$ if \[A^2=AB+2A,\] where $B$ is the following matrix. If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$. (a) \[B=\begin{bmatrix} -1 & 1 & -1 \\ 0 &-1 &0 \\ 1 & 2 & […]
• Overall Fraction of Defective Smartphones of Three Factories A certain model of smartphone is manufactured by three factories A, B, and C. Factories A, B, and C produce $60\%$, $25\%$, and $15\%$ of the smartphones, respectively. Suppose that their defective rates are $5\%$, $2\%$, and $7\%$, respectively. Determine the overall fraction of […]
• A Group Homomorphism that Factors though Another Group Let $G, H, K$ be groups. Let $f:G\to K$ be a group homomorphism and let $\pi:G\to H$ be a surjective group homomorphism such that the kernel of $\pi$ is included in the kernel of $f$: $\ker(\pi) \subset \ker(f)$. Define a map $\bar{f}:H\to K$ as follows. For each […]
• Successful Probability of a Communication Network Diagram Consider the network diagram in the figure. The diagram consists of five links and each of them fails to communicate with probability $p$. Answer the following questions about this network. (1) Determine the probability that there exists at least one path from A to B where every […]
• Vector Form for the General Solution of a System of Linear Equations Solve the following system of linear equations by transforming its augmented matrix to reduced echelon form (Gauss-Jordan elimination). Find the vector form for the general […]
• Application of Field Extension to Linear Combination Consider the cubic polynomial $f(x)=x^3-x+1$ in $\Q[x]$. Let $\alpha$ be any real root of $f(x)$. Then prove that $\sqrt{2}$ can not be written as a linear combination of $1, \alpha, \alpha^2$ with coefficients in $\Q$.   Proof. We first prove that the polynomial […]
• Find a Matrix so that a Given Subset is the Null Space of the Matrix, hence it’s a Subspace Let $W$ be the subset of $\R^3$ defined by \[W=\left \{ \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\in \R^3 \quad \middle| \quad 5x_1-2x_2+x_3=0 \right \}.\] Exhibit a $1\times 3$ matrix $A$ such that $W=\calN(A)$, the null space of $A$. […]