91Ó°ÊÓ

Given a homogeneous system of linear equations, if the system is overdetermined, what are the possibilities as to the number of solutions? Explain.

Is the transpose of an elementary matrix an elementary matrix of the same type? Is the product of two elementary matrices an elementary matrix?

Let $$A=\left(\begin{array}{ll} O & I \\ B & O \end{array}\right)$$ where all four submatrices are \(k \times k\). Determine \(A^{2}\) and \(A^{4}\)

Let \(A \mathbf{x}=\mathbf{b}\) be a linear system whose augmented matrix ( \(A | \mathbf{b}\) ) has reduced row echelon form $$\left(\begin{array}{ccccc|r} 1 & 2 & 0 & 3 & 1 & -2 \\ 0 & 0 & 1 & 2 & 4 & 5 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right)$$ (a) Find all solutions to the system. (b) If $$\mathbf{a}_{1}=\left(\begin{array}{l} 1 \\ 1 \\ 3 \\ 4 \end{array}\right) \quad \text { and } \quad \mathbf{a}_{3}=\left(\begin{array}{r} 2 \\ -1 \\ 1 \\ 3 \end{array}\right)$$ determine b.

Let \(U\) and \(R\) be \(n \times n\) upper triangular matrices and set \(T=U R .\) Show that \(T\) is also upper triangular and that \(t_{j j}=u_{j j} r_{j j}\) for \(j=1, \ldots, n\)

Given a nonhomogeneous system of linear equations, if the system is underdetermined, what are the possibilities as to the number of solutions? Explain.

Let \(A\) be an \(m \times n\) matrix. Explain why the matrix multiplications \(A^{T} A\) and \(A A^{T}\) are possible.

Prove that if \(A\) is nonsingular then \(A^{T}\) is nonsingular and $$\left(A^{T}\right)^{-1}=\left(A^{-1}\right)^{T}$$ Hint: \((A B)^{T}=B^{T} A^{T}\)

A matrix \(A\) is said to be skew symmetric if \(A^{T}=-A .\) Show that if a matrix is skew symmetric, then its diagonal entries must all be 0.

Let \(A\) be a nonsingular \(n \times n\) matrix. Use mathematical induction to prove that \(A^{m}\) is nonsingular and $$\left(A^{m}\right)^{-1}=\left(A^{-1}\right)^{m}$$ for \(m=1,2,3, \ldots\)