91Ó°ÊÓ

Verify by direct multiplication that the given matrices are inverses of one another. $$A=\left[\begin{array}{ll} 2 & -1 \\ 3 & -1 \end{array}\right], A^{-1}=\left[\begin{array}{ll} -1 & 1 \\ -3 & 2 \end{array}\right]$$

Verify that the given triple of real numbers is a solution to the given system. (2,-3,1); $$\begin{aligned}x_{1}+x_{2}-2 x_{3} &=-3 \\\3 x_{1}-x_{2}-7 x_{3} &=2 \\ x_{1}+x_{2}+x_{3} &=0 \\\2 x_{1}+2 x_{2}-4 x_{3} &=-6\end{aligned}$$

Verify that for all values of \(t\) $$(1-t, 2+3 t, 3-2 t)$$ is a solution to the linear system $$\begin{aligned}x_{1}+x_{2}+x_{3} &=6. \\\x_{1}-x_{2}-2 x_{3} &=-7. \\\5 x_{1}+x_{2}-x_{3} &=4.\end{aligned}$$

Express the matrix \(A\) as a product of elementary matrices. $$A=\left[\begin{array}{ll}1 & 2 \\\1 & 3\end{array}\right]$$

Determine \(A^{-1},\) if possible, using the Gauss-Jordan method. If \(A^{-1}\) exists, check your answer by verifying that \(A A^{-1}=I_{n}\) $$A=\left[\begin{array}{rrr} 1 & -1 & 2 \\ 2 & 1 & 11 \\ 4 & -3 & 10 \end{array}\right]$$

Express the matrix \(A\) as a product of elementary matrices. $$A=\left[\begin{array}{rr}4 & -5 \\\1 & 4\end{array}\right]$$

Determine elementary matrices \(E_{1}, E_{2}, \ldots, E_{k}\) that reduce $$A=\left[\begin{array}{rr}2 & -1 \\\1 & 3\end{array}\right]$$ to reduced row-echelon form. Verify by direct multiplication that \(E_{1} E_{2} \ldots E_{k} A=I_{2}\).

use elementary row operations to reduce the given matrix to row-echelon form, and hence determine the rank of each matrix. $$\left[\begin{array}{cccc} 2 & -2 & -1 & 3 \\ 3 & -2 & 3 & 1 \\ 1 & -1 & 1 & 0 \\ 2 & -1 & 2 & 2 \end{array}\right]$$.

Consider the \(m \times n\) homogeneous system of linear equations $$A \mathbf{x}=\mathbf{0}$$ (a) If \(\mathbf{x}=\left[x_{1} x_{2} \ldots x_{n}\right]^{T}\) and \(\mathbf{y}=\left[\begin{array}{lll}y_{1} & y_{2} & \dots & y_{n}\end{array}\right]^{T}\) are solutions to \((2.3 .4),\) show that $$\mathbf{z}=\mathbf{x}+\mathbf{y} \text { and } \mathbf{w}=c \mathbf{x}$$ are also solutions, where \(c\) is an arbitrary scalar. (b) Is the result of (a) true when \(x\) and \(y\) are solutions to the nonhomogeneous system \(A \mathbf{x}=\mathbf{b} ?\) Explain.

The Pauli spin matrices \(\sigma_{1}, \sigma_{2},\) and \(\sigma_{3}\) are defined by $$ \sigma_{1}=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right], \sigma_{2}=\left[\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right] $$ and $$ \sigma_{3}=\left[\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right] $$ Verify that they satisfy $$ \sigma_{1} \sigma_{2}=i \sigma_{3}, \sigma_{2} \sigma_{3}=i \sigma_{1}, \sigma_{3} \sigma_{1}=i \sigma_{2} $$ If \(A\) and \(B\) are \(n \times n\) matrices, we define their commutator, denoted \([A, B],\) by $$ [A, B]=A B-B A $$ Thus, \([A, B]=0\) if and only if \(A\) and \(B\) commute. That is, \(A B=B A .\) Problems \(19-22\) require the commutator.