In: Math
For each of the statements (i)—(iii), state whether it is true or false (i) Every system of linear equations has at least one solution. (ii) A system of four linear equations with three variables always has infinitely many solutions. (iii) One can determine whether two straight lines in R 3 intersect by solving an appropriate system of linear equations. (iv) Given any matrix A, then its reduced row echelon form is not unique. (v) Every elementary row operation can be undone by an(other) elementary row operation. (vi) One of the elementary row operations is to delete a row (i) A system of 3 linear equations with 4 variables cannot have a unique solution. (ii) It is possible to obtain two different reduced row echelon matrices from a given matrix by using two different sequences of elementary row operations. (iii) Elementary row operations on an augmented matrix do not change the solution set of the associated system of linear equations. (i) The matrix ? ? 1 0 a 0 100 0 0 0 7 ? ? is invertible for any value of a. (ii) If a system of linear equations with a square coefficient matrix A has infinitely many solutions, then det(A) = 0. (iii) Elementary row operations do not change the determinant of matrices. (i) A 2 × 2-matrix can have three distinct eigenvalues. (ii) If 0 is an eigenvalue of a square matrix A, then A is not invertible. (iii) Every 3 × 3-matrix can be diagonalized using elementary row operations.