91Ó°ÊÓ

Let \\[ A=\left(\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right) \\] Show that if \(d=a_{11} a_{22}-a_{21} a_{12} \neq 0,\) then \\[ A^{-1}=\frac{1}{d}\left(\begin{array}{rr} a_{22} & -a_{12} \\ -a_{21} & a_{11} \end{array}\right) \\]

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

Let \(A\) be a nonsingular matrix. Show that \(A^{-1}\) is also nonsingular and \(\left(A^{-1}\right)^{-1}=A\).

Prove that if \(A\) is nonsingular, then \(A^{T}\) is nonsingular and \\[ \left(A^{T}\right)^{-1}=\left(A^{-1}\right)^{T} \\] \(\left[\text {Hint: }(A B)^{T}=B^{T} A^{T} .\right]\)

Let \(A\) be an \(n \times n\) matrix and let \(\mathbf{x}\) and \(\mathbf{y}\) be vectors in \(\mathbb{R}^{n} .\) Show that if \(A \mathbf{x}=A \mathbf{y}\) and \(\mathbf{x} \neq \mathbf{y},\) then the matrix \(A\) must be singular.

Let \(U\) be an \(n \times n\) upper triangular matrix with nonzero diagonal entries. (a) Explain why \(U\) must be nonsingular. (b) Explain why \(U^{-1}\) must be upper triangular.

Nitric acid is prepared commercially by a series of three chemical reactions. In the first reaction, nitro\(\operatorname{gen}\left(\mathrm{N}_{2}\right)\) is combined with hydrogen \(\left(\mathrm{H}_{2}\right)\) to form ammonia \(\left(\mathrm{NH}_{3}\right) .\) Next, the ammonia is combined with oxygen \(\left(\mathrm{O}_{2}\right)\) to form nitrogen dioxide \(\left(\mathrm{NO}_{2}\right)\) and water. Finally, the \(\mathrm{NO}_{2}\) reacts with some of the water to form nitric acid (HNO \(_{3}\) ) and nitric oxide (NO). The amounts of each of the components of these reactions are measured in moles (a standard unit of measurement for chemical reactions). How many moles of nitrogen, hydrogen, and oxygen are necessary to produce 8 moles of nitric acid?

Show that if \(A\) is a symmetric nonsingular matrix, then \(A^{-1}\) is also symmetric.

Prove that if \(A\) is row equivalent to \(B\), then \(B\) is row equivalent to \(A\)

Let \(A\) be an idempotent matrix. (a) Show that \(I-A\) is also idempotent. (b) Show that \(I+A\) is nonsingular and \((I+A)^{-1}=I-\frac{1}{2} A\)