(Solved): For the linear system \( A \vec{x}=\vec{b}, A \) is a \( 4 \times 4 \) matrix with rank 3 and the ...

For the linear system \( A \vec{x}=\vec{b}, A \) is a \( 4 \times 4 \) matrix with rank 3 and the augmented matrix \( [A \mid \vec{b}] \) is a \( 4 \times 5 \) matrix with rank 4 . How many solutions does this system of equations have? A. The system has a unique solution. B. The system has no solution. C. The system has a one-parameter family of solutions. D. The system has a two-parameter family of solutions. E. The system has a three-parameter family of solutions. Find the rank of \( \left[\begin{array}{rrr}1 & -2 & 3 \\ 2 & -4 & 6 \\ -1 & 2 & -3\end{array}\right] \) A. 0 B. 2 C. 3 D. 4 E. 1 Let \( A \vec{x}=\vec{b} \) be a system of 5 linear equations in 6 unknowns. Which one of the following statements \( < \) strong \( > \) MUST \( \) be \( < \) strong \( > \) false \( \) ? A. The system might have a one-parameter family of solutions. B. The system might have no solution. C. The system might have a unique solution. D. The system might have a two-parameter family of solutions. E. The system might have a three-parameter family of solutions. Which one of the following statements is false? A. If \( A \) is any \( 5 \times 5 \) matrix with \( \operatorname{rank}(A)=4 \), then the linear system \( A \vec{x}=\vec{b} \), with \( \vec{b} \neq \overrightarrow{0} \), must have infinitely many solutions. B. If \( A \) is any \( 5 \times 5 \) invertible matrix, then the linear system \( A \vec{x}=\overrightarrow{0} \) has a unique solution. C. Any homogeneous system of 5 linear equations in 7 unknowns always has a nontrivial solution. D. If \( A \) is any \( 5 \times 5 \) matrix with \( \operatorname{rank}(A)=5 \), then \( A \) is invertible. Reset Selection