91Ó°ÊÓ

Prove or give a counterexample to the following statement: If the coefficient matrix of a system of \(m\) linear equations in \(n\) unknowns has rank \(m\), then the system has a solution.

In the closed model of Leontief with food, clothing, and housing as the basic industries, suppose that the input-output matrix is $$ A=\left(\begin{array}{rrr} \frac{7}{16} & \frac{1}{2} & \frac{3}{16} \\ \frac{5}{16} & \frac{1}{6} & \frac{5}{16} \\ \frac{1}{4} & \frac{1}{3} & \frac{1}{2} \end{array}\right) \text {. } $$ At what ratio must the farmer, tailor, and carpenter produce in order for equilibrium to be attained?

Let \(B^{\prime}\) and \(D^{\prime}\) be \(m \times n\) matrices, and let \(B\) and \(D\) be \((m+1) \times(n+1)\) matrices respectively defined by $$ B=\left(\begin{array}{c|ccc} 1 & 0 & \cdots & 0 \\ \hline 0 & & & \\ \vdots & & B^{\prime} & \\ 0 & & & \end{array}\right) \text { and } D=\left(\begin{array}{c|ccc} 1 & 0 & \cdots & 0 \\ \hline 0 & & & \\ \vdots & & D^{\prime} & \\ 0 & & & \end{array}\right) $$ Prove that if \(B^{\prime}\) can be transformed into \(D^{\prime}\) by an elementary row [column] operation, then \(B\) can be transformed into \(D\) by an elementary row [column] operation.

Let \(A\) be an \(m \times n\) matrix. Prove that there exists a sequence of elementary row operations of types 1 and 3 that transforms \(A\) into an upper triangular matrix.

Let \(\mathrm{T}, \mathrm{U}: \mathrm{V} \rightarrow \mathrm{W}\) be linear transformations. (a) Prove that \(R(T+U) \subseteq R(T)+R(U)\). (See the definition of the sum of subsets of a vector space on page 22.) (b) Prove that if \(\mathrm{W}\) is finite-dimensional, then \(\operatorname{rank}(\mathrm{T}+\mathrm{U}) \leq \operatorname{rank}(\mathrm{T})+\) \(\operatorname{rank}(U)\). (c) Deduce from (b) that \(\operatorname{rank}(A+B) \leq \operatorname{rank}(A)+\operatorname{rank}(B)\) for any \(m \times n\) matrices \(A\) and \(B\).

Let \(A\) be an \(m \times n\) matrix and \(B\) be an \(n \times p\) matrix. Prove that \(A B\) can be written as a sum of \(n\) matrices of rank at most one.

Let \(A\) be an \(m \times n\) matrix with rank \(m\) and \(B\) be an \(n \times p\) matrix with rank \(n\). Determine the rank of \(A B\). Justify your answer.

Let $$ A=\left(\begin{array}{rrrrr} 1 & 0 & -1 & 2 & 1 \\ -1 & 1 & 3 & -1 & 0 \\ -2 & 1 & 4 & -1 & 3 \\ 3 & -1 & -5 & 1 & -6 \end{array}\right) $$ (a) Find a \(5 \times 5\) matrix \(M\) with rank 2 such that \(A M=O\), where \(O\) is the \(4 \times 5\) zero matrix. (b) Suppose that \(B\) is a \(5 \times 5\) matrix such that \(A B=O\). Prove that \(\operatorname{rank}(B) \leq 2\).

Let \(A\) be an \(m \times n\) matrix with rank \(m\). Prove that there exists an \(n \times m\) matrix \(B\) such that \(A B=I_{m}\).