MATH 54

Midterm 1 | Spring '16 | Yuan

1. The following matrix is the augmented matrix of a system of linear equations. Write down the solution set of each as a linear combination of vectors if it is consistent, explain the reason if it is inconsistent.
3-variable system: $\left( \begin{array}{c c c c} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 3 \end{array} \right)$

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The following matrix is the augmented matrix of a system of linear equations. Write down the solution set of each as a linear combination of vectors if it is consistent, explain the reason if it is inconsistent.
3-variable system: $\left( \begin{array}{c c c c} 1 & 0 & 2 & 0 \\ 0 & 1 & 3 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right)$

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The following matrix is the augmented matrix of a system of linear equations. Write down the solution set of each as a linear combination of vectors if it is consistent, explain the reason if it is inconsistent.
4-variable system: $\left( \begin{array}{c c c c c} 1 & 0 & 2 & 3 & 4 \\ 0 & 1 & -5 & -6 & -7 \end{array} \right)$

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2. (5 points) Solve the system $\begin{cases} x_1 + 3x_2 + 5x_3 &= 1 \\ -x_1 + 2x_2 + 2x_3 &= 2 \\ x_1 + x_3 &= 3 \end{cases}$
(The final result could be either inconsistent or an experience of a general solution.)

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(5 points) Compute the inverse of the matrix $\left( \begin{array}{c c c} 3 & 0 & 1 \\ 0 & 1 & -2 \\ 1 & 0 & 0 \end{array} \right)$

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4. (5 points) Let $A$ be the matrix $A = \left( \begin{array}{c c c} 3 & 4 & a \\ 1 & 0 & 0 \\ 2 & 1 & 5 \end{array} \right)$
Note that $a$ is entry of $A$. Find all values of $a$ such that the linear transformation $L(\vec{x}) = A\vec{x}$ from $\mathbb{R}^3$ to $\mathbb{R}^3$ is not onto. (Note “onto" and “surjective" have the same meaning.)

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(5 points) Let $A, B, C$ be $2 \times 2$ matrices, and let $M = ABC$ be their products. Assume that the second row of $A$ is $(1 \; 2 \; 3)$, the second column of $C$ is $\left( \begin{array}{c} 1 \\ -2 \\ -1 \end{array} \right)$ and that $B = \left( \begin{array}{c c c} 0 & 0 & 1 \\ 0 & -2 & 0 \\ 1 & 1 & 0 \end{array} \right)$ What is the $(2,2)$-entry of $M$? (Note: the $(2,2)$-entry means the entry in the second row and second column.)

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(5 points) Let $A = (\vec{a}_1 \; \vec{a}_2 \; \vec{a}_3)$ be a $3 \times 3$ matrix (so that $\vec{a}_1, \vec{a}_2, \vec{a}_3$ are the common of $A$.) Let $B = ( \vec{a}_1 + \vec{a}_2 + \vec{a}_3 \quad \vec{a}_1 + 2\vec{a}_2 + 3\vec{a}_3 \quad \vec{a}_1 + 4\vec{a}_2 + 9\vec{a}_3)$ be the $3 \times 3$ matrix whose columns are $\vec{a}_1 + \vec{a}_2 + \vec{a}_3 , \quad \vec{a}_1 + 2\vec{a}_2 + 3\vec{a}_3 , \quad \vec{a}_1 + 4\vec{a}_2 + 9\vec{a}_3$ Assume that $\det (A) = 2016$. Find $\det (B)$.

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