Chapter 2

Let \(A=\left[\begin{array}{rrrr}-2 & 4 & 2 & 6 \\ -1 & -1 & 5 & 0\end{array}\right], B=\left[\begin{array}{rr}-3 & 0 \\ 2 & 2 \\ 1 & -3 \\\ 0 & 1\end{array}\right], C=\) \(\left[\begin{array}{r}-5 \\ -6 \\ 3 \\ 1\end{array}\right] .\) Compute the given expression, if possible. $$A^{T}-5 B.$$

Write all \(3 \times 3\) elementary matrices and their inverses.

Use part (c) of the Invertible Matrix Theorem to prove that if \(A\) is an invertible matrix and \(B\) and \(C\) are matrices of the same size as \(A\) such that \(A B=A C\), then \(B=C .\) IHint: Consider \(A B-A C=0.1.\)

Use Gaussian elimination to determine the solution set to the given system. $$\begin{aligned} x_{1}-5 x_{2} &=3 \\ 3 x_{1}-9 x_{2} &=15 \end{aligned}$$

Let $$A=\left[\begin{array}{ccc} -2 & 6 & 1 \\ -1 & 0 & -3 \end{array}\right], B=\left[\begin{array}{ccc} 2 & 1 & -1 \\ 0 & 4 & -4 \end{array}\right]$$,$$C=\left[\begin{array}{ll} 1+i & 2+i \\ 3+i & 4+i \\ 5+i & 6+i \end{array}\right], D=\left[\begin{array}{lll} 4 & 0 & 1 \\ 1 & 2 & 5 \\ 3 & 1 & 2 \end{array}\right]$$ $$E=\left[\begin{array}{rrr} 2 & -5 & -2 \\ 1 & 1 & 3 \\ 4 & -2 & -3 \end{array}\right], F=\left[\begin{array}{ccc} 6 & 2-3 i & i \\ 1+i & -2 i & 0 \\ -1 & 5+2 i & 3 \end{array}\right]$$ In these problems, \(i\) denotes \(\sqrt{-1}\) (a) \(5 A\) (b) \(-3 B\) (c) \(i C\) (d) \(2 A-B\) (e) \(A+3 C^{T}\) (f) \(3 D-2 E\) (g) \(D+E+F\) (h) the matrix \(G\) such that \(2 A+3 B-2 G=5(A+B)\) (i) the matrix \(H\) such that \(D+2 F+H=4 E\) (j) the matrix \(K\) such that \(K^{T}+3 A-2 B=0_{2 \times 3}\)

Verify by direct multiplication that the given matrices are inverses of one another. $$A=\left[\begin{array}{ll} 4 & 9 \\ 3 & 7 \end{array}\right], A^{-1}=\left[\begin{array}{rr} 7 & -9 \\ -3 & 4 \end{array}\right]$$

Verify that the given triple of real numbers is a solution to the given system. $$(1,-1,2);$$ $$\begin{aligned}2 x_{1}-3 x_{2}+4 x_{3} &=13 \\\x_{1}+x_{2}-x_{3} &=-2 \\ 5 x_{1}+4 x_{2}+x_{3} &=3\end{aligned}$$

determine whether the given matrices are in reduced row-echelon form, row- echelon form but not reduced row-echelon form, or neither. $$\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]$$.

Determine elementary matrices that reduce the given matrix to row-echelon form. $$\left[\begin{array}{rr}-4 & -1 \\\0 & 3 \\\\-3 & 7\end{array}\right]$$

Let \(A=\left[\begin{array}{rrrr}-2 & 4 & 2 & 6 \\ -1 & -1 & 5 & 0\end{array}\right], B=\left[\begin{array}{rr}-3 & 0 \\ 2 & 2 \\ 1 & -3 \\\ 0 & 1\end{array}\right], C=\) \(\left[\begin{array}{r}-5 \\ -6 \\ 3 \\ 1\end{array}\right] .\) Compute the given expression, if possible. $$C^{T} B.$$

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

Let $$A=\left[\begin{array}{ccc} -2 & 6 & 1 \\ -1 & 0 & -3 \end{array}\right], B=\left[\begin{array}{ccc} 2 & 1 & -1 \\ 0 & 4 & -4 \end{array}\right]$$,$$C=\left[\begin{array}{ll} 1+i & 2+i \\ 3+i & 4+i \\ 5+i & 6+i \end{array}\right], D=\left[\begin{array}{lll} 4 & 0 & 1 \\ 1 & 2 & 5 \\ 3 & 1 & 2 \end{array}\right]$$ $$E=\left[\begin{array}{rrr} 2 & -5 & -2 \\ 1 & 1 & 3 \\ 4 & -2 & -3 \end{array}\right], F=\left[\begin{array}{ccc} 6 & 2-3 i & i \\ 1+i & -2 i & 0 \\ -1 & 5+2 i & 3 \end{array}\right]$$ In these problems, \(i\) denotes \(\sqrt{-1}\) Compute each of the following: (a) \(-D\) (b) \(4 B^{T}\) (c) \(-2 A^{T}+C\) (d) \(5 E+D\) (e) \(4 A^{T}-2 B^{T}+i C\) (f) \(4 E-3 D^{T}\) \((g)(1-6 i) F+i D\) (h) the matrix \(G\) such that \(2 A-B+(1-i) C^{T}=\) \(G+A-B\) (i) the matrix \(H\) such that \(3 D-3 E+6 I_{3}-2 H=0_{3}\) (j) the matrix \(K\) such that \(K^{T}+F^{T}=D^{T}+E^{T}\)

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

determine whether the given matrices are in reduced row-echelon form, row- echelon form but not reduced row-echelon form, or neither. $$\left[\begin{array}{ll} 1 & 1 \\ 0 & 0 \end{array}\right]$$.

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

$$^{165}=\left[\begin{array}{c}-1^{-1}-\frac{1}{-3} \\ -\frac{5}{-1} \\ -1 \\\ 6 \\ -\frac{4}{4}\end{array}\right]$$ (a) \(b_{12}, b_{33}, b_{41}, b_{43}, b_{51},\) and \(b_{52}\) (b) all pairs \((i, j)\) such that \(b_{i j}=-1\)

Determine elementary matrices that reduce the given matrix to row-echelon form. $$\left[\begin{array}{rr}3 & 5 \\\1 & -2\end{array}\right]$$

Let $$A=\left[\begin{array}{rrr} 1 & -1 & 2 \\ 3 & 1 & 4 \end{array}\right], B=\left[\begin{array}{rrr} 2 & -1 & 3 \\ 5 & 1 & 2 \\ 4 & 6 & -2 \end{array}\right]$$ $$C=\left[\begin{array}{r} 1 \\ -1 \\ 2 \end{array}\right], D=\left[\begin{array}{lll} 2 & -2 & 3 \end{array}\right]$$ $$E=\left[\begin{array}{cc} 2-i & 1+i \\ -i & 2+4 i \end{array}\right], F=\left[\begin{array}{cc} i & 1-3 i \\ 0 & 4+i \end{array}\right]$$ In these problems, \(i\) denotes \(\sqrt{-1}\) For each item, decide whether or not the given expression is defined. For each item that is defined, compute the result. (a) \(A B\) (b) \(B C\) (c) \(C A\) (d) \(A^{T} E\) (e) \(C D\) (f) \(C^{T} A^{T}\) (g) \(F^{2}\) (h) \(B D^{T}\) (i) \(A^{T} A\) (j) \(F E\)

determine whether the given matrices are in reduced row-echelon form, row- echelon form but not reduced row-echelon form, or neither. $$\left[\begin{array}{llll} 1 & 0 & 2 & 5 \\ 1 & 0 & 0 & 2 \\ 0 & 1 & 1 & 0 \end{array}\right]$$.

Verify by direct multiplication that the given matrices are inverses of one another. \(A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right], A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{rr}d & -b \\ -c & a\end{array}\right]\) provided \(a d-b c \neq 0\)

Use Gaussian elimination to determine the solution set to the given system. $$\begin{array}{r} 7 x_{1}-3 x_{2}=5 \\ 14 x_{1}-6 x_{2}=10 \end{array}$$

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

Write the matrix with the given elements In each case, specify the dimensions of the matrix. $$a_{11}=1, a_{21}=-1, a_{12}=5, a_{22}=3$$

Let \(A=\left[\begin{array}{rrrr}-2 & 4 & 2 & 6 \\ -1 & -1 & 5 & 0\end{array}\right], B=\left[\begin{array}{rr}-3 & 0 \\ 2 & 2 \\ 1 & -3 \\\ 0 & 1\end{array}\right], C=\) \(\left[\begin{array}{r}-5 \\ -6 \\ 3 \\ 1\end{array}\right] .\) Compute the given expression, if possible. $$-4 A-B^{T}.$$

Use the equivalence of (a) and (e) in the Invertible Matrix Theorem to prove that if \(A\) and \(B\) are invertible \(n \times n\) matrices, then so is \(A B.\)

determine whether the given matrices are in reduced row-echelon form, row- echelon form but not reduced row-echelon form, or neither. $$\left[\begin{array}{rrrr} 1 & 0 & -1 & 0 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right]$$.

Use Gaussian elimination to determine the solution set to the given system. $$\begin{aligned} x_{1}+2 x_{2}+x_{3} &=1 \\ 3 x_{1}+5 x_{2}+x_{3} &=3 \\ 2 x_{1}+6 x_{2}+7 x_{3} &=1 \end{aligned}$$

Verify by direct multiplication that the given matrices are inverses of one another. $$A=\left[\begin{array}{lll} 3 & 5 & 1 \\ 1 & 2 & 1 \\ 2 & 6 & 7 \end{array}\right], A^{-1}=\left[\begin{array}{rrr} 8 & -29 & 3 \\ -5 & 19 & -2 \\ 2 & -8 & 1 \end{array}\right]$$

Verify that for all values of \(s\) and \(t\) $$(s, s-2 t, 2 s+3 t, t)$$ is a solution to the linear system $$\begin{aligned}x_{1}+x_{2}-x_{3}+5 x_{4} &=0 \\\2 x_{2}-x_{3}+7 x_{4} &=0 \\\4 x_{1}+2 x_{2}-3 x_{3}+13 x_{4} &=0 \end{aligned}$$

Determine elementary matrices that reduce the given matrix to row-echelon form. $$\left[\begin{array}{rrr}3 & -1 & 4 \\\2 & 1 & 3 \\\1 & 3 & 2\end{array}\right]$$

Let \(A=\left[\begin{array}{rrrr}-2 & 4 & 2 & 6 \\ -1 & -1 & 5 & 0\end{array}\right], B=\left[\begin{array}{rr}-3 & 0 \\ 2 & 2 \\ 1 & -3 \\\ 0 & 1\end{array}\right], C=\) \(\left[\begin{array}{r}-5 \\ -6 \\ 3 \\ 1\end{array}\right] .\) Compute the given expression, if possible. $$A B \text { and } \operatorname{tr}(A B).$$

Use the equivalence of (a) and (c) in the Invertible Matrix Theorem to prove that if \(A\) and \(B\) are invertible \(n \times n\) matrices, then so is \(A B.\)

$$\begin{aligned} &\text { Let } A=\left[\begin{array}{rrrr} -3 & 2 & 7 & -1 \\ 6 & 0 & -3 & -5 \end{array}\right], B=\left[\begin{array}{rr} -2 & 8 \\ 8 & -3 \\ -1 & -9 \\ 0 & 2 \end{array}\right]\\\ &\text { and } C=\left[\begin{array}{rr} -6 & 1 \\ 1 & 5 \end{array}\right] . \text { Compute } A B C \text { and } C A B \end{aligned}$$

determine whether the given matrices are in reduced row-echelon form, row- echelon form but not reduced row-echelon form, or neither. $$\left[\begin{array}{llll} 1 & 0 & 1 & 2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]$$.

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

Make a sketch in the \(x y\) -plane in order to determine the number of solutions to the given linear system. $$\begin{aligned}&3 x-4 y=12\\\&6 x-8 y=24\end{aligned}$$

Use Gaussian elimination to determine the solution set to the given system. $$\begin{aligned} &3 x_{1}-x_{2} \quad=\quad 1\\\ &2 x_{1}+x_{2}+5 x_{3}=4\\\ &7 x_{1}-5 x_{2}-8 x_{3}=-3 \end{aligned}$$

Let \(A=\left[\begin{array}{rrrr}-2 & 4 & 2 & 6 \\ -1 & -1 & 5 & 0\end{array}\right], B=\left[\begin{array}{rr}-3 & 0 \\ 2 & 2 \\ 1 & -3 \\\ 0 & 1\end{array}\right], C=\) \(\left[\begin{array}{r}-5 \\ -6 \\ 3 \\ 1\end{array}\right] .\) Compute the given expression, if possible. $$(A C)(A C)^{T}.$$

Determine elementary matrices that reduce the given matrix to row-echelon form. $$\left[\begin{array}{llll}1 & 2 & 3 & 4 \\\2 & 3 & 4 & 5 \\\3 & 4 & 5 & 6\end{array}\right]$$

Determine \(A\) c by computing an appropriate linear combination of the column vectors of \(A\). $$A=\left[\begin{array}{rl} 1 & 3 \\ -5 & 4 \end{array}\right], \mathbf{c}=\left[\begin{array}{r} 6 \\ -2 \end{array}\right]$$

determine whether the given matrices are in reduced row-echelon form, row- echelon form but not reduced row-echelon form, or neither. $$\left[\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$.

Make a sketch in the \(x y\) -plane in order to determine the number of solutions to the given linear system. $$\begin{aligned}&2 x+3 y=1\\\&2 x+3 y=2\end{aligned}$$

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

Use Gaussian elimination to determine the solution set to the given system. $$\begin{aligned} 3 x_{1}+5 x_{2}-x_{3} &=14 \\ x_{1}+2 x_{2}+x_{3} &=3 \\ 2 x_{1}+5 x_{2}+6 x_{3} &=2 \end{aligned}$$

Write the matrix with the given elements In each case, specify the dimensions of the matrix. $$\begin{array}{l} a_{11}=1, a_{31}=2, a_{42}=-1, a_{32}=7, a_{13}=-2 \\ a_{23}=0, a_{33}=4, a_{21}=3, a_{41}=-4, a_{12}=-3 \\ a_{22}=6, a_{43}=5 \end{array}$$

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

Determine \(A\) c by computing an appropriate linear combination of the column vectors of \(A\). $$A=\left[\begin{array}{rrr} 3 & -1 & 4 \\ 2 & 1 & 5 \\ 7 & -6 & 3 \end{array}\right], \mathbf{c}=\left[\begin{array}{r} 2 \\ 3 \\ -4 \end{array}\right]$$

determine whether the given matrices are in reduced row-echelon form, row- echelon form but not reduced row-echelon form, or neither. $$\left[\begin{array}{llll} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$$.

Make a sketch in the \(x y\) -plane in order to determine the number of solutions to the given linear system. $$\begin{array}{r}x+4 y=8 \\\3 x+y=3\end{array}$$

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