For each of the statements (i)—(iii), state whether it is trueor false (i) Every system of linear equations has at least onesolution. (ii) A system of four linear equations with threevariables always has infinitely many solutions. (iii) One candetermine whether two straight lines in R 3 intersect by solving anappropriate system of linear equations. (iv) Given any matrix A,then its reduced row echelon form is not unique. (v) Everyelementary row operation can be undone by an(other) elementary rowoperation. (vi) One of the elementary row operations is to delete arow (i) A system of 3 linear equations with 4 variables cannot havea unique solution. (ii) It is possible to obtain two differentreduced row echelon matrices from a given matrix by using twodifferent sequences of elementary row operations. (iii) Elementaryrow operations on an augmented matrix do not change the solutionset 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) Ifa system of linear equations with a square coefficient matrix A hasinfinitely many solutions, then det(A) = 0. (iii) Elementary rowoperations do not change the determinant of matrices. (i) A 2 ×2-matrix can have three distinct eigenvalues. (ii) If 0 is aneigenvalue of a square matrix A, then A is not invertible. (iii)Every 3 × 3-matrix can be diagonalized using elementary rowoperations.
