Linear equation forms questions and answers

The reduced row echelon form of a system of linear equations is given.Write the system of equations corresponding to the given matrix. Use x, y. or x, y, z. or \(x_1, x_2, x_3, x_4\) as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.
\(\begin{bmatrix}1 & 0 & 0 & 0 & 1\\ 0 &1 & 0 &0 & 2 \\ 0 & 0 & 1 & 2 & 3 \end{bmatrix}\)

The row echelon form of a system of linear equations is given.
a) Write the system of equations corresponding to the given matrix. Use x, y. or x, y, z. or \(x_1,x_2,x_3,x_4\) as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.
\(\begin{bmatrix}1 & 2 & -1 & 0 \\ 0 & 1 & -1 & 1 \\ 0 & 0 & 0 & 2 \end{bmatrix}\)

The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. \(\begin{bmatrix} {1}&{0}&-{1}&-{2}&-{3}&{1}\\{0}&{1}&{2}&{3}&{4}&{0}\end{bmatrix}\)

The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. \(\begin{bmatrix}{1}&{0}&-{1}&-{2}&-{3}&{0}\\{0}&{1}&{2}&{3}&{4}&{0}\end{bmatrix}\)

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system of linear equations is consistent and, if so, find its general solution. \(\begin{bmatrix} {0}&{0}&{1}&-{3}&{0}&{2}&{0}\\ {0}&{0}&{0}&{0}&{1}&-{1}&{0}\\ {0}&{0}&{0}&{0}&{0}&{0}&{0}\\ \end{bmatrix}\)

The given matrix is the augmented matrix for a system of linear equations \(\Bigg[ \begin{array}{} 1&0&-1&0&-1&-2&0\\ 0&1&2&0&1&2&0 \\ 0&0&0&1&1&1&0\\ \end{array} \Bigg]\)

Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given row echelon form. Solve the system by back substitution. Assume that the variables are named \(\displaystyle{x}_{{1}},{x}_{{2}},\ldots\) from left to right.
\(\begin{bmatrix}1 & 1&-3&2&1 \\0&1&4&0&3\\0&0&0&1&2 \end{bmatrix}\)

Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row echelon form. Solve the system. Assume that the variables are named \(x_1, x_2,...x,...\) ,… from left to right. 

Let \(AX = B\) be a system of linear equations, where A is an \(m\times nm\times n\) matrix, X is an n-vector, and \(BB\) is an m-vector. Assume that there is one solution \(X=X0\). Show that every solution is of the form \(X0+Y\), where Y is a solution of the homogeneous system \(AY = 0\), and conversely any vector of the form \(X0+Y\) is a solution.

The reduced row echelon form of a system of linear equations is given.Write the system of equations corresponding to the given matrix. Use x, y; or x, y, z; or \(x_1,x_2,x_3,x_4\) as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution. \(\left[\begin{array}{cccc|c} 1&0&0&4&2\\0&1&1&3&3\\0&0&0&0&0 \end{array}\right]\)

The reduced row echelon form of a system of linear equations is given.Write the system of equations corresponding to the given matrix. Use x, y; or x, y, z; or \(x_1,x_2,x_3,x_4\) as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution. \(\left[\begin{array}{ccc|c}{1} & {0} & {4} & {4} \\ {0} & {1} & {3} & {2} \\ {0} & {0} & {0} & {0}\end{array}\right]\)

The row echelon form of a system of linear equations is given.
(a)Write the system of equations corresponding to the given matrix.Usex,y;orx,y,z;or \(x_1,x_2,x_3,x_4\) asvariables.
(b)Determine whether the system is consistent or inconsistent. If it is consistent, give the solution. \(\left[\begin{array}{cc|c} 1&2&5\\0&1&-1 \end{array}\right]\)