Chapter 1 - Systems of Linear Equations - 1.2 Gaussian Elimination and Gauss-Jordan Elimination - 1.2 Exercises - Page 24: 57

a) true b) true c) false d) true

a) If $A$ is any matrix with $m$ rows and $n$ columns, then $A$ is said to be a $m\times n$ matrix. So, a $6\times 3$ matrix has $6$ rows. b) True. Every matrix is row-equivalent to a matrix in row-echelon form as every row-reduced echelon form of a matrix is obtained from an augmented matrix using a series of elementary row operations. c) False. The system is not necessary consistent. Clearly $[1$ $0$ $0$ $0$ $0]$ is the first row, and the last row can be of the form $[0$ $0$ $0$ $0$ $0]$; the system then would be consistent. d) True. Since we have $4$ linear equations with $6$ variables, the number of equations is less than the number of variables, which implies that some variables will be free variables. Hence, we will get infinitely many solutions.