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MATH1061/1002/1021 (ND) MATH1061 Canvas Quiz 6

Single choice

Row echelon forms of the augmented matrices for three systems of linear equations in \(x\), \(y\) and \(z\) are given below. For each augmented matrix, \(A\), \(B\) and \(C\), which of the following options correctly describes the number of solutions to the corresponding system of linear equations? \[ \underbrace { \begin{bmatrix} 1 & 0 & 0 & \bigm | & -2 \\ 0 & 1 & 0 & \bigm | & 1 \\ 0 & 0 & 0 & \bigm | & -2 \end{bmatrix} }_{A} \qquad \underbrace { \begin{bmatrix} 1 & 0 & 0 & \bigm | & 0 \\ 0 & 1 & 0 & \bigm | & 2 \\ 0 & 0 & 1 & \bigm | & -1 \end{bmatrix} }_{B} \qquad \underbrace { \begin{bmatrix} 1 & 0 & 0 & \bigm | & 1 \\ 0 & 1 & 0 & \bigm | & -3 \\ 0 & 0 & 0 & \bigm | & 0 \end{bmatrix} }_{C} \]

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First, restating the problem in my own words helps focus: we have three augmented matrices A, B, and C, each representing a system of linear equations in x, y, z. We must determine the number of solutions for each system based on their row echelon forms. Option A describes A as inconsistent. Looking at A, the last row is [0 0 0 | -2]. This corresponds to the equation 0x + 0y + 0z = -2, which is impossible. Suc......Login to view full explanation

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