Quiz: Linear Equations and Matrix Entreis

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• Find a Basis For the Null Space of a Given $2\times 3$ Matrix Let \[A=\begin{bmatrix} 1 & 1 & 0 \\ 1 &1 &0 \end{bmatrix}\] be a matrix. Find a basis of the null space of the matrix $A$. (Remark: a null space is also called a kernel.)   Solution. The null space $\calN(A)$ of the matrix $A$ is by […]
• Use 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 $A_1, A_2$ […]
• Find Values of $a$ so that Augmented Matrix Represents a Consistent System Suppose that the following matrix $A$ is the augmented matrix for a system of linear equations. \[A= \left[\begin{array}{rrr|r} 1 & 2 & 3 & 4 \\ 2 &-1 & -2 & a^2 \\ -1 & -7 & -11 & a \end{array} \right],\] where $a$ is a real number. Determine all the […]
• Quiz 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 […]
• Quiz 4: Inverse Matrix/ Nonsingular Matrix Satisfying a Relation (a) Find the inverse matrix of \[A=\begin{bmatrix} 1 & 0 & 1 \\ 1 &0 &0 \\ 2 & 1 & 1 \end{bmatrix}\] if it exists. If you think there is no inverse matrix of $A$, then give a reason. (b) Find a nonsingular $2\times 2$ matrix $A$ such that \[A^3=A^2B-3A^2,\] where […]
• Summary: Possibilities for the Solution Set of a System of Linear Equations In this post, we summarize theorems about the possibilities for the solution set of a system of linear equations and solve the following problems. Determine all possibilities for the solution set of the system of linear equations described below. (a) A homogeneous system of $3$ […]
• Possibilities of the Number of Solutions of a Homogeneous System of Linear Equations Here is a very short true or false problem. Select either True or False. Then click "Finish quiz" button. You will be able to see an explanation of the solution by clicking "View questions" button.  
• True or False: $(A-B)(A+B)=A^2-B^2$ for Matrices $A$ and $B$ Let $A$ and $B$ be $2\times 2$ matrices. Prove or find a counterexample for the statement that $(A-B)(A+B)=A^2-B^2$.   Hint. In general, matrix multiplication is not commutative: $AB$ and $BA$ might be different. Solution. Let us calculate $(A-B)(A+B)$ as […]