91Ó°ÊÓ

Label the following statements as true or false. (a) An elementary matrix is always square. (b) The only entries of an elementary matrix are zeros and ones. (c) The \(n \times n\) identity matrix is an elementary matrix. (d) The product of two \(n \times n\) elementary matrices is an elementary matrix. (e) The inverse of an elementary matrix is an elementary matrix. (f) The sum of two \(n \times n\) elementary matrices is an elementary matrix. (g) The transpose of an elementary matrix is an elementary matrix. (h) If \(B\) is a matrix that can be obtained by performing an elementary row operation on a matrix \(A\), then \(B\) can also be obtained by performing an elementary column operation on \(A\). (i) If \(B\) is a matrix that can be obtained by performing an elementary row operation on a matrix \(A\), then \(A\) can be obtained by performing an elementary row operation on \(B\).

Let $$ A=\left(\begin{array}{rrr} 1 & 2 & 3 \\ 1 & 0 & 1 \\ 1 & -1 & 1 \end{array}\right), B=\left(\begin{array}{rrr} 1 & 0 & 3 \\ 1 & -2 & 1 \\ 1 & -3 & 1 \end{array}\right), \text { and } C=\left(\begin{array}{rrr} 1 & 0 & 3 \\ 0 & -2 & -2 \\ 1 & -3 & 1 \end{array}\right) \text {. } $$ Find an elementary operation that transforms \(A\) into \(B\) and an elementary operation that transforms \(B\) into \(C\). By means of several additional operations, transform \(C\) into \(I_{3}\).

Suppose that the augmented matrix of a system \(A x=b\) is transformed into a matrix \(\left(A^{\prime} \mid b^{\prime}\right)\) in reduced row echelon form by a finite sequence of elementary row operations. (a) Prove that rank \(\left(A^{\prime}\right) \neq \operatorname{rank}\left(A^{\prime} \mid b^{\prime}\right)\) if and only if \(\left(A^{\prime} \mid b^{\prime}\right)\) contains a row in which the only nonzero entry lies in the last column. (b) Deduce that \(A x=b\) is consistent if and only if \(\left(A^{\prime} \mid b^{\prime}\right)\) contains no row in which the only nonzero entry lies in the last column.

Prove that for any \(m \times n\) matrix \(A, \operatorname{rank}(A)=0\) if and only if \(A\) is the zero matrix.

Prove that \(E\) is an elementary matrix if and only if \(E^{t}\) is.

Let the reduced row echelon form of \(A\) be $$ \left(\begin{array}{rrrrr} 1 & 0 & 2 & 0 & -2 \\ 0 & 1 & -5 & 0 & -3 \\ 0 & 0 & 0 & 1 & 6 \end{array}\right) $$ Determine \(A\) if the first, second, and fourth columns of \(A\) are $$ \left(\begin{array}{r} 1 \\ -1 \\ 3 \end{array}\right), \quad\left(\begin{array}{r} 0 \\ -1 \\ 1 \end{array}\right), \quad \text { and } \quad\left(\begin{array}{r} 1 \\ -2 \\ 0 \end{array}\right) $$ respectively.Let the reduced row echelon form of \(A\) be $$ \left(\begin{array}{rrrrr} 1 & 0 & 2 & 0 & -2 \\ 0 & 1 & -5 & 0 & -3 \\ 0 & 0 & 0 & 1 & 6 \end{array}\right) $$ Determine \(A\) if the first, second, and fourth columns of \(A\) are $$ \left(\begin{array}{r} 1 \\ -1 \\ 3 \end{array}\right), \quad\left(\begin{array}{r} 0 \\ -1 \\ 1 \end{array}\right), \quad \text { and } \quad\left(\begin{array}{r} 1 \\ -2 \\ 0 \end{array}\right) $$ respectively.Let the reduced row echelon form of \(A\) be $$ \left(\begin{array}{rrrrr} 1 & 0 & 2 & 0 & -2 \\ 0 & 1 & -5 & 0 & -3 \\ 0 & 0 & 0 & 1 & 6 \end{array}\right) $$ Determine \(A\) if the first, second, and fourth columns of \(A\) are $$ \left(\begin{array}{r} 1 \\ -1 \\ 3 \end{array}\right), \quad\left(\begin{array}{r} 0 \\ -1 \\ 1 \end{array}\right), \quad \text { and } \quad\left(\begin{array}{r} 1 \\ -2 \\ 0 \end{array}\right) $$ respectively.Let the reduced row echelon form of \(A\) be $$ \left(\begin{array}{rrrrr} 1 & 0 & 2 & 0 & -2 \\ 0 & 1 & -5 & 0 & -3 \\ 0 & 0 & 0 & 1 & 6 \end{array}\right) $$ Determine \(A\) if the first, second, and fourth columns of \(A\) are $$ \left(\begin{array}{r} 1 \\ -1 \\ 3 \end{array}\right), \quad\left(\begin{array}{r} 0 \\ -1 \\ 1 \end{array}\right), \quad \text { and } \quad\left(\begin{array}{r} 1 \\ -2 \\ 0 \end{array}\right) $$ respectively.

Prove that if a matrix \(Q\) can be obtained from a matrix \(P\) by an elementary row operation, then \(P\) can be obtained from \(Q\) by an elementary row operation of the same type. Hint: Treat each type of elementary row operation separately.

Let \(A\) be an \(m \times n\) matrix. Prove that if \(c\) is any nonzero scalar, then \(\operatorname{rank}(c A)=\operatorname{rank}(A)\).

Prove that the system of linear equations \(A x=b\) has a solution if and only if \(b \in \mathrm{R}\left(\mathrm{L}_{A}\right)\). Visit goo.gl/JfwjBa for a solution.

Prove that any elementary row [column] operation of type 1 can be obtained by a succession of three elementary row [column] operations of type 3 followed by one elementary row [column] operation of type 2 . Visit goo.gl/oNJBFz for a solution.