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Math exam problems and solutions at Harvard University


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  • Find Inverse Matrices Using Adjoint MatricesFind Inverse Matrices Using Adjoint Matrices Let $A$ be an $n\times n$ matrix. The $(i, j)$ cofactor $C_{ij}$ of $A$ is defined to be \[C_{ij}=(-1)^{ij}\det(M_{ij}),\] where $M_{ij}$ is the $(i,j)$ minor matrix obtained from $A$ removing the $i$-th row and $j$-th column. Then consider the $n\times n$ matrix […]
  • Quiz 6. Determine Vectors in Null Space, Range / Find a Basis of Null SpaceQuiz 6. Determine Vectors in Null Space, Range / Find a Basis of Null Space (a) Let $A=\begin{bmatrix} 1 & 2 & 1 \\ 3 &6 &4 \end{bmatrix}$ and let \[\mathbf{a}=\begin{bmatrix} -3 \\ 1 \\ 1 \end{bmatrix}, \qquad \mathbf{b}=\begin{bmatrix} -2 \\ 1 \\ 0 \end{bmatrix}, \qquad \mathbf{c}=\begin{bmatrix} 1 \\ 1 […]
  • The Subspace of Linear Combinations whose Sums of Coefficients are zeroThe Subspace of Linear Combinations whose Sums of Coefficients are zero Let $V$ be a vector space over a scalar field $K$. Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ be vectors in $V$ and consider the subset \[W=\{a_1\mathbf{v}_1+a_2\mathbf{v}_2+\cdots+ a_k\mathbf{v}_k \mid a_1, a_2, \dots, a_k \in K \text{ and } […]
  • Top 10 Popular Math Problems in 2016-2017Top 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 […]
  • Every Ideal of the Direct Product of Rings is the Direct Product of IdealsEvery Ideal of the Direct Product of Rings is the Direct Product of Ideals Let $R$ and $S$ be rings with $1\neq 0$. Prove that every ideal of the direct product $R\times S$ is of the form $I\times J$, where $I$ is an ideal of $R$, and $J$ is an ideal of $S$.   Proof. Let $K$ be an ideal of the direct product $R\times […]
  • Find All the Eigenvalues of $A^k$ from Eigenvalues of $A$Find All the Eigenvalues of $A^k$ from Eigenvalues of $A$ Let $A$ be $n\times n$ matrix and let $\lambda_1, \lambda_2, \dots, \lambda_n$ be all the eigenvalues of $A$. (Some of them may be the same.) For each positive integer $k$, prove that $\lambda_1^k, \lambda_2^k, \dots, \lambda_n^k$ are all the eigenvalues of […]
  • Use Cramer’s Rule to Solve a $2\times 2$ System of Linear EquationsUse Cramer’s Rule to Solve a $2\times 2$ System of Linear Equations Use Cramer's rule to solve the system of linear equations \begin{align*} 3x_1-2x_2&=5\\ 7x_1+4x_2&=-1. \end{align*}   Solution. Let \[A=[A_1, A_2]=\begin{bmatrix} 3 & -2\\ 7& 4 \end{bmatrix},\] be the coefficient matrix of the system, where $A_1, A_2$ […]
  • Example of an Infinite Group Whose Elements Have Finite OrdersExample of an Infinite Group Whose Elements Have Finite Orders Is it possible that each element of an infinite group has a finite order? If so, give an example. Otherwise, prove the non-existence of such a group.   Solution. We give an example of a group of infinite order each of whose elements has a finite order. Consider […]

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