For each of the statements (i)—(iii), state whether it is true or false (i) Every system...

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.

Solutions

Expert Solution

Every system of linear equations has at least one solution. False. The system may be inconsistent.
A system of four linear equations with three variables always has infinitely many solutions. False. These linear equations may be linearly dependent.
One can determine whether two straight lines in R³ intersect by solving an appropriate system of linear equations. True.
Given any matrix A, its reduced row echelon form is not unique. False.
Every elementary row operation can be undone by an(other) elementary row operation. True
One of the elementary row operations is to delete a row. False.
A system of 3 linear equations with 4 variables cannot have a unique solution. True, because there is a free variable.
It is possible to obtain two different reduced row echelon matrices from a given matrix by using two different sequences of elementary row operations. False.
Elementary row operations on an augmented matrix do not change the solution set of the associated system of linear equations. True.
The matrix ? ? 1 0 a 0 100 0 0 0 7 ? ? is invertible for any value of a. Cannot understand the ? marks.
If a system of linear equations with a square coefficient matrix A has infinitely many solutions, then det(A) = 0. True.
Elementary row operations do not change the determinant of matrices. False.
A 2 × 2-matrix can have three distinct eigenvalues. False
If 0 is an eigenvalue of a square matrix A, then A is not invertible. True.
Every 3 × 3-matrix can be diagonalized using elementary row operations. False. It can be diagonalized only if it has 3 distinct and linearly independent eigenvectors.

milcah answered 1 year ago

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