Chapter 2

Determine \(A^{-1},\) if possible, using the Gauss-Jordan method. If \(A^{-1}\) exists, check your answer by verifying that \(A A^{-1}=I_{n}\) $$A=\left[\begin{array}{ccc} 1 & i & 2 \\ 1+i & -1 & 2 i \\ 2 & 2 i & 5 \end{array}\right]$$

Write the column vectors and row vectors of the given matrix. $$A=\left[\begin{array}{rrr} 1 & 3 & -4 \\ -1 & -2 & 5 \\ 2 & 6 & 7 \end{array}\right]$$

An \(n \times n\) matrix \(A\) is called nilpotent if \(A^{p}=0\) for some positive integer \(p .\) Show that the given matrix is nilpotent. $$A=\left[\begin{array}{lll}0 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{array}\right]$$

Determine a Type 3 lower triangular elementary ma\(\operatorname{trix} E_{1}\) that reduces \(A=\left[\begin{array}{rr}3 & -2 \\ -1 & 5\end{array}\right]\) to upper triangular form. Use Equation \((2.7 .3)\) to determine \(L\) and verify Equation (2.7.2).

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

Write the system of equations with the given coefficient matrix and right-hand side vector. $$A=\left[\begin{array}{rr}0 & -3 \\\2 & -7 \\\5 & 5\end{array}\right], \mathbf{b}=\left[\begin{array}{r}-1 \\\6 \\\7\end{array}\right].$$

Use Gauss-Jordan elimination to determine the solution set to the given system. $$\begin{aligned} x_{1} &-2 x_{3}=-3 \\ 3 x_{1}-2 x_{2}-4 x_{3} &=-9 \\ x_{1}-4 x_{2}+2 x_{3} &=-3 \end{aligned}$$

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

Write the column vectors and row vectors of the given matrix. $$A=\left[\begin{array}{ccc} 2 & 10 & 6 \\ 5 & -1 & 3 \end{array}\right]$$

Determine the LU factorization of the given matrix. Verify your answer by computing the product \(L U\). $$A=\left[\begin{array}{ll}2 & 3 \\\5 & 1\end{array}\right]$$

use elementary row operations to reduce the given matrix to row-echelon form, and hence determine the rank of each matrix. $$\left[\begin{array}{cccc} 2 & -1 & 3 & 4 \\ 1 & -2 & 1 & 3 \\ 1 & -5 & 0 & 5 \end{array}\right]$$.

If \(A=\left[\begin{array}{rl}x & 1 \\ -2 & y\end{array}\right],\) determine all values of \(x\) and \(y\) for which \(A^{2}=A\)

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.

Use Gauss-Jordan elimination to determine the solution set to the given system. $$\begin{aligned} 2 x_{1}-x_{2}+3 x_{3}-x_{4} &=3 \\ 3 x_{1}+2 x_{2}+x_{3}-5 x_{4} &=-6 \\ x_{1}-2 x_{2}+3 x_{3}+x_{4} &=6 \end{aligned}$$

Determine \(A^{-1},\) if possible, using the Gauss-Jordan method. If \(A^{-1}\) exists, check your answer by verifying that \(A A^{-1}=I_{n}\) $$A=\left[\begin{array}{rrrr} 1 & -1 & 2 & 3 \\ 2 & 0 & 3 & -4 \\ 3 & -1 & 7 & 8 \\ 1 & 0 & 3 & 5 \end{array}\right]$$

If \(\mathbf{a}_{1}=\left[\begin{array}{lll}1 & 2\end{array}\right], \mathbf{a}_{2}=\left[\begin{array}{ll}3 & 4\end{array}\right],\) and \(\mathbf{a}_{3}=\left[\begin{array}{ll}5 & 1\end{array}\right],\) write the matrix $$A=\left[\begin{array}{l} \mathbf{a}_{1} \\ \mathbf{a}_{2} \\ \mathbf{a}_{3} \end{array}\right]$$ and determine the column vectors of \(A\)

Let \(A(t)=\left[\begin{array}{cc}e^{-3 t} & -\sec ^{2} t \\ 2 t^{3} & \cos t \\\ 6 \ln t & 36-5 t\end{array}\right]\) and \(B(t)=\left[\begin{array}{cc}-7 & t^{2} \\ 6-t & 3 t^{3}+6 t^{2} \\ 1+t & \cos (\pi t / 2) \\ e^{t} & 1-t^{3}\end{array}\right] .\) Compute the given expression, if possible. $$\int_{0}^{1} B(t) d t$$

Determine the LU factorization of the given matrix. Verify your answer by computing the product \(L U\). $$A=\left[\begin{array}{ll}3 & 1 \\\5 & 2\end{array}\right]$$

use elementary row operations to reduce the given matrix to row-echelon form, and hence determine the rank of each matrix. $$\left[\begin{array}{lllll} 2 & 1 & 3 & 4 & 2 \\ 1 & 0 & 2 & 1 & 3 \\ 2 & 3 & 1 & 5 & 7 \end{array}\right]$$.

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.

Write the vector formulation for the given system of differential equations. $$x_{1}^{\prime}=-4 x_{1}+3 x_{2}+4 t, x_{2}^{\prime}=6 x_{1}-4 x_{2}+t^{2}$$

Use Gauss-Jordan elimination to determine the solution set to the given system. $$\begin{aligned} &x_{1}+x_{2}+x_{3}-x_{4}=4\\\ &x_{1}-x_{2}-x_{3}-x_{4}=2\\\ &x_{1}+x_{2}-x_{3}+x_{4}=-2\\\ &x_{1}-x_{2}+x_{3}+x_{4}=-8 \end{aligned}$$

Determine \(A^{-1},\) if possible, using the Gauss-Jordan method. If \(A^{-1}\) exists, check your answer by verifying that \(A A^{-1}=I_{n}\) $$A=\left[\begin{array}{rrrr} 0 & -2 & -1 & -3 \\ 2 & 0 & 2 & 1 \\ 1 & -2 & 0 & 2 \\ 3 & -1 & -2 & 0 \end{array}\right]$$

If \(\mathbf{a}_{1}=\left[\begin{array}{lllll}-2 & 0 & 4 & -1 & -1\end{array}\right]\) and \(\mathbf{a}_{2}=\left[\begin{array}{lllll}9 & -4 & -4 & 0 & 8\end{array}\right],\) write the matrix $$A=\left[\begin{array}{l} \mathbf{a}_{1} \\ \mathbf{a}_{2} \end{array}\right]$$ and determine the column vectors of \(A .\)

Let \(A(t)=\left[\begin{array}{cc}e^{-3 t} & -\sec ^{2} t \\ 2 t^{3} & \cos t \\\ 6 \ln t & 36-5 t\end{array}\right]\) and \(B(t)=\left[\begin{array}{cc}-7 & t^{2} \\ 6-t & 3 t^{3}+6 t^{2} \\ 1+t & \cos (\pi t / 2) \\ e^{t} & 1-t^{3}\end{array}\right] .\) Compute the given expression, if possible. $$t^{3} \cdot A(t)-\sin t \cdot B(t)$$

Determine the LU factorization of the given matrix. Verify your answer by computing the product \(L U\). $$A=\left[\begin{array}{rrr}3 & -1 & 2 \\\6 & -1 & 1 \\\\-3 & 5 & 2\end{array}\right]$$

use elementary row operations to reduce the given matrix to row-echelon form, and hence determine the rank of each matrix. $$\left[\begin{array}{rrrr} 4 & 7 & 4 & 7 \\ 3 & 5 & 3 & 5 \\ 2 & -2 & 2 & -2 \\ 5 & -2 & 5 & -2 \end{array}\right]$$.

$$\text { If } A=\left[\begin{array}{rr} 1 & -1 \\ 2 & 1 \end{array}\right], B=\left[\begin{array}{ll} 3 & 1 \\ 4 & 2 \end{array}\right], \text { find }[A, B]$$

Write the vector formulation for the given system of differential equations. $$x_{1}^{\prime}=-4 x_{1}+3 x_{2}+4 t, x_{2}^{\prime}=6 x_{1}-4 x_{2}+t^{2}$$

Use Gauss-Jordan elimination to determine the solution set to the given system. $$\begin{aligned} 2 x_{1}-x_{2}+3 x_{3}+x_{4}-x_{5} &=11 \\ x_{1}-3 x_{2}-2 x_{3}-x_{4}-2 x_{5} &=2 \\ 3 x_{1}+x_{2}-2 x_{3}-x_{4}+x_{5} &=-2 \\ x_{1}+2 x_{2}+x_{3}+2 x_{4}+3 x_{5} &=-3 \\ 5 x_{1}-3 x_{2}-3 x_{3}+x_{4}+2 x_{5} &=2 \end{aligned}$$

If $$\mathbf{b}_{1}=\left[\begin{array}{r}-2 \\\\-6 \\\3 \\\\-1 \\\\-2\end{array}\right] \quad \text { and } \quad \mathbf{b}_{2}=\left[\begin{array}{r}-4 \\\\-6 \\\0 \\\0 \\\1\end{array}\right]$$ write the matrix \(B=\left[\mathbf{b}_{1}, \mathbf{b}_{2}\right],\) and determine the row vectors of \(B\).

Determine the LU factorization of the given matrix. Verify your answer by computing the product \(L U\). $$A=\left[\begin{array}{rrr}5 & 2 & 1 \\\\-10 & -2 & 3 \\\15 & 2 & -3\end{array}\right]$$

Let \(A(t)=\left[\begin{array}{cc}e^{-3 t} & -\sec ^{2} t \\ 2 t^{3} & \cos t \\\ 6 \ln t & 36-5 t\end{array}\right]\) and \(B(t)=\left[\begin{array}{cc}-7 & t^{2} \\ 6-t & 3 t^{3}+6 t^{2} \\ 1+t & \cos (\pi t / 2) \\ e^{t} & 1-t^{3}\end{array}\right] .\) Compute the given expression, if possible. $$B^{\prime}(t)-e^{t} A(t)$$

Reduce the given matrix to reduced rowechelon form and hence determine the rank of each matrix. $$\left[\begin{array}{cc} -4 & 2 \\ -6 & 3 \end{array}\right]$$.

If \(A_{1}=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right], A_{2}=\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right],\) and \(A_{3}=\) \(\left[\begin{array}{ll}0 & 0 \\ 1 & 0\end{array}\right],\) compute all of the commutators \(\left[A_{i}, A_{j}\right]\) and determine which of the matrices commute.

Write the vector formulation for the given system of differential equations. $$x_{1}^{\prime}=e^{2 t} x_{2}, x_{2}^{\prime}+(\sin t) x_{1}=1$$

Determine the solution set to the sys\(\operatorname{tem} A \mathbf{x}=\mathbf{b}\) for the given coefficient matrix \(A\) and right-hand side vector b. $$A=\left[\begin{array}{rrr} 1 & -3 & 1 \\ 5 & -4 & 1 \\ 2 & 4 & -3 \end{array}\right], \mathbf{b}=\left[\begin{array}{r} 8 \\ 15 \\ -4 \end{array}\right]$$

Let \(A=\left[\begin{array}{ccc}-1 & -2 & 3 \\ -1 & 1 & 1 \\ -1 & -2 & -1\end{array}\right] .\) Find the third column vector of \(A^{-1}\) without determining the other columns of the inverse matrix.

If $$\begin{aligned}\mathbf{b}_{1}=&\left[\begin{array}{r}2 \\\\-1 \\\4\end{array}\right], \quad \mathbf{b}_{2}=\left[\begin{array}{r}5 \\\7 \\\\-6\end{array}\right] \\\\\mathbf{b}_{3}=\left[\begin{array}{l}0 \\ 0 \\\0\end{array}\right], & \mathbf{b}_{4}=\left[\begin{array}{l}1 \\\2 \\\3\end{array}\right]\end{aligned}$$ write the matrix \(B=\left[\mathbf{b}_{1}, \mathbf{b}_{2}, \mathbf{b}_{3}, \mathbf{b}_{4}\right],\) and determine the row vectors of \(B\)

Determine the solution set to the given linear system of equations. $$\begin{aligned} x_{1}+5 x_{2}+2 x_{3} &=-6, \\ 4 x_{2}-7 x_{3} &=2, \\ 5 x_{3} &=0. \end{aligned}$$

Determine the LU factorization of the given matrix. Verify your answer by computing the product \(L U\). $$A=\left[\begin{array}{rrrr}1 & -1 & 2 & 3 \\\2 & 0 & 3 & -4 \\\3 & -1 & 7 & 8 \\\1 & 3 & 4 & 5 \end{array}\right]$$

Reduce the given matrix to reduced rowechelon form and hence determine the rank of each matrix. $$\left[\begin{array}{rr} 3 & 2 \\ 1 & -1 \end{array}\right]$$.

$$\begin{aligned} &\text { If } A_{1}=\frac{1}{2}\left[\begin{array}{ll} 0 & i \\ i & 0 \end{array}\right], A_{2}=\frac{1}{2}\left[\begin{array}{rr} 0 & -1 \\ 1 & 0 \end{array}\right], \text { and }\\\ &A_{3}=\frac{1}{2}\left[\begin{array}{cc} i & 0 \\ 0 & -i \end{array}\right], \text { verify that }\\\ &\left[A_{1}, A_{2}\right]=A_{3},\left[A_{2}, A_{3}\right]=A_{1},\left[A_{3}, A_{1}\right]=A_{2} \end{aligned}$$

Write the vector formulation for the given system of differential equations. $$\begin{array}{l}x_{1}^{\prime}=(-\sin t) x_{2}+x_{3}+t, x_{2}^{\prime}=-e^{t} x_{1}+t^{2} x_{3}+t^{3} \\\x_{3}^{\prime}=-t x_{1}+t^{2} x_{2}+1\end{array}$$

Determine the solution set to the sys\(\operatorname{tem} A \mathbf{x}=\mathbf{b}\) for the given coefficient matrix \(A\) and right-hand side vector b. $$A=\left[\begin{array}{rrr} 1 & 0 & 5 \\ 3 & -2 & 11 \\ 2 & -2 & 6 \end{array}\right], \mathbf{b}=\left[\begin{array}{l} 0 \\ 2 \\ 2 \end{array}\right]$$

If \(\mathbf{a}_{1}, \mathbf{a}_{2}, \ldots, \mathbf{a}_{p}\) are each column \(q\) -vectors, what are the dimensions of the matrix that has \(\mathbf{a}_{1}, \mathbf{a}_{2}, \ldots, \mathbf{a}_{p}\) as its column vectors?

Determine the solution set to the given linear system of equations. $$\begin{aligned} 5 x_{1}-x_{2}+2 x_{3} &=7, \\ -2 x_{1}+6 x_{2}+9 x_{3} &=0, \\ -7 x_{1}+5 x_{2}-3 x_{3} &=-7. \end{aligned}$$

Determine the LU factorization of the given matrix. Verify your answer by computing the product \(L U\). $$A=\left[\begin{array}{rrrr}2 & -3 & 1 & 2 \\\4 & -1 & 1 & 1 \\\\-8 & 2 & 2 & -5 \\\6 & 1 & 5 & 2 \end{array}\right]$$

Reduce the given matrix to reduced rowechelon form and hence determine the rank of each matrix. $$\left[\begin{array}{rrr} 3 & 7 & 10 \\ 2 & 3 & -1 \\ 1 & 2 & 1 \end{array}\right]$$.

Verify that the given vector function \(\mathbf{x}\) defines a solution to \(\mathbf{x}^{\prime}=A \mathbf{x}+\mathbf{b}\) for the given \(A\) and \(\mathbf{b}.\) $$\begin{aligned}&\mathbf{x}(t)=\left[\begin{array}{c}e^{4 t} \\\\-2 e^{4 t}\end{array}\right], A=\left[\begin{array}{rr}2 & -1 \\\\-2 & 3\end{array}\right]\\\&\mathbf{b}(t)=\left[\begin{array}{l}0 \\\0\end{array}\right]\end{aligned}.$$