?Use the inversion algorithm to find the inverse of the matrix (if the inverse exists)

In Exercises 1–2, determine whether the given matrix is elementary. a. \(\left[\begin{array}{rr}1 & 0 \\-5 & 1\end{array}\right]\) B. \(\left[\begin{array}{rr}-5 & 1 \\1 & 0\end{array}\right]\) C. \(\left[\begin{array}{lll}1 & 1 & 0 \\0 & 0 & 1 \\0 & 0 & 0\end{array}\right]\) D. \(\left[\begin{array}{llll}2 & 0 & 0 & 2 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{array}\right]\) Equation Transcription: [] [] [] [] Text Transcription: [-5 1 1 0] [ 1 0 -5 1] [ 0 0 0 0 0 1 1 1 0] [0 0 0 1 0 0 1 0 0 1 0 0 2 0 0 2]

Determine whether the given matrix is elementary. a. \(\left[\begin{array}{rr}1 & 0 \\0 & \sqrt{3}\end{array}\right]\) b. \(\left[\begin{array}{lll}0 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & 0\end{array}\right]\) c. \(\left[\begin{array}{lll}1 & 0 & 0 \\0 & 1 & 9 \\0 & 0 & 1\end{array}\right]\) d. \(\left[\begin{array}{rrr}-1 & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0\end{array}\right]\) Equation Transcription: [] [] [] [] Text Transcription: [0 3 1 0] [1 0 0 0 1 0 0 0 1] [0 0 1 0 1 9 1 0 0] [ 0 1 0 0 0 1 -1 0 0]

Find a row operation and the corresponding elementary matrix that will restore the given elementary matrix to the identity matrix. a. \(\left[\begin{array}{rr}1 & -3 \\0 & 1\end{array}\right]\) b. \(\left[\begin{array}{rrr}-7 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{array}\right]\) c. \(\left[\begin{array}{rrr}1 & 0 & 0 \\0 & 1 & 0 \\-5 & 0 & 1\end{array}\right]\) d. \(\left[\begin{array}{llll}0 & 0 & 1 & 0 \\0 & 1 & 0 & 0 \\1 & 0 & 0 & 0 \\0 & 0 & 0 & 1\end{array}\right]\) Equation Transcription: [] [] [] [] Text Transcription: [0 1 1 -3] [ 0 0 1 0 1 0 -7 0 0] [-5 0 1 0 1 9 1 0 0] [0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0]

Find a row operation and the corresponding elementary matrix that will restore the given elementary matrix to the identity matrix. a. \(\left[\begin{array}{rr}1 & 0 \\-3 & 1\end{array}\right]\) b. \(\left[\begin{array}{rr}1 & 0 \\-3 & 1\end{array}\right]\) c. \(\left[\begin{array}{rr}1 & 0 \\-3 & 1\end{array}\right]\) D. \(\left[\begin{array}{llll}1 & 0 & -\frac{1}{7} & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{array}\right]\) Equation Transcription: [] [] [] [] Text Transcription: [-3 1 1 0] [ 0 0 3 0 1 0 1 0 0] [1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1] [0 0 0 1 0 0 1 0 0 1 0 0 1 0 -1/7 0]

An elementary matrix \(E\) and a matrix \(A\) are given. Identify the row operation corresponding to \(E\) and verify that the product \(EA\) results from applying the row operation to \(A\). a. \(E=\left[\begin{array}{ll}0 & 1 \\1 & 0\end{array}\right], \quad A=\left[\begin{array}{rrrr}-1 & -2 & 5 & -1 \\3 & -6 & -6 & -6\end{array}\right]\) b.. \(E=\left[\begin{array}{rrr}1 & 0 & 0 \\0 & 1 & 0 \\0 & -3 & 1\end{array}\right], \quad A=\left[\begin{array}{rrrrr}2 & -1 & 0 & -4 & -4 \\1 & -3 & -1 & 5 & 3 \\2 & 0 & 1 & 3 & -1\end{array}\right]\) c. \(E=\left[\begin{array}{lll}1 & 0 & 4 \\0 & 1 & 0 \\0 & 0 & 1\end{array}\right], \quad A=\left[\begin{array}{ll}1 & 4 \\2 & 5 \\3 & 6\end{array}\right]\) Equation Transcription: [], [] [], [] [], [] Text Transcription: E A EA E =[1 0 0 1], A =[ 3 -6 -6 -6 -1 -2 5 -1] E =[ 0 -3 1 0 1 01 0 0], A =[ 2 0 1 3 -1 1 -3 -1 5 3 2 -1 0 -4 -4] E =[ 0 0 1 0 1 0 1 0 4], A =[ 3 6 2 5 1 4]

An elementary matrix \(E\) and a matrix \(A\) are given. Identify the row operation corresponding to \(E\) and verify that the product \(EA\) results from applying the row operation to \(A\). a. \(E=\left[\begin{array}{rr}-6 & 0 \\0 & 1\end{array}\right], \quad A=\left[\begin{array}{rrrr}-1 & -2 & 5 & -1 \\3 & -6 & -6 & -6\end{array}\right]\) b. \(E=\left[\begin{array}{rrr}1 & 0 & 0 \\-4 & 1 & 0 \\0 & 0 & 1\end{array}\right], \quad A=\left[\begin{array}{rrrrr}2 & -1 & 0 & -4 & -4 \\1 & -3 & -1 & 5 & 3 \\2 & 0 & 1 & 3 & -1\end{array}\right]\) C. \(E=\left[\begin{array}{lll}1 & 0 & 0 \\0 & 5 & 0 \\0 & 0 & 1\end{array}\right], \quad A=\left[\begin{array}{ll}1 & 4 \\2 & 5 \\3 & 6\end{array}\right]\) Equation Transcription: [], [] [], [] [], [] Text Transcription: E A EA E =[ 0 1-6 0], A =[ 3 -6 -6 -6 -1 -2 5 -1] E =[ 0 0 1 -4 1 0 1 0 0], A =[ 2 0 1 3 -1 1 -3 -1 5 32 -1 0 -4 -4] E =[ 0 0 1 0 5 01 0 0], A =[ 3 6 2 5 1 4]

Use the following matrices and find an elementary matrix \(E\) that satisfies the stated equation. \(A=\left[\begin{array}{rrr}3 & 4 & 1 \\2 & -7 & -1 \\8 & 1 & 5\end{array}\right]\), \(B=\left[\begin{array}{rrr}8 & 1 & 5 \\2 & -7 & -1 \\3 & 4 & 1\end{array}\right]\) \(C=\left[\begin{array}{rrr}3 & 4 & 1 \\2 & -7 & -1 \\2 & -7 & 3\end{array}\right]\), \(D=\left[\begin{array}{rrr}8 & 1 & 5 \\-6 & 21 & 3 \\3 & 4 & 1\end{array}\right]\) \(F=\left[\begin{array}{lll}8 & 1 & 5 \\8 & 1 & 1 \\3 & 4 & 1\end{array}\right]\) a. \(EA = B\) b. \(EB = A\) c. \(EA = C\) d. \(EC = A\) Equation Transcription: [], [] [], [] [] Text Transcription: E A =[8 1 5 2 -7 -1 3 4 1], B =[3 4 1 2 -7 -1 8 1 5] C =[2 -7 3 2 -7 -1 3 4 1], D =[ 3 4 1 -6 21 3 8 1 5] D =[ 3 4 1 8 1 1 8 1 5] EA = B EB = A EA = C EC = A

Use the following matrices and find an elementary matrix \(E\) that satisfies the stated equation. \(A=\left[\begin{array}{rrr}3 & 4 & 1 \\2 & -7 & -1 \\8 & 1 & 5\end{array}\right]\), \(B=\left[\begin{array}{rrr}8 & 1 & 5 \\2 & -7 & -1 \\3 & 4 & 1\end{array}\right]\) \(C=\left[\begin{array}{rrr}3 & 4 & 1 \\2 & -7 & -1 \\2 & -7 & 3\end{array}\right]\), \(D=\left[\begin{array}{rrr}8 & 1 & 5 \\-6 & 21 & 3 \\3 & 4 & 1\end{array}\right]\) \(F=\left[\begin{array}{lll}8 & 1 & 5 \\8 & 1 & 1 \\3 & 4 & 1\end{array}\right]\) a. \(EB = D\) b. \(ED = B\) c. \(EB = F\) d. \(EF = B\) Equation Transcription: [], [] [], [] [] Text Transcription: E A =[8 1 5 2 -7 -1 3 4 1], B =[3 4 1 2 -7 -1 8 1 5] C =[2 -7 3 2 -7 -1 3 4 1], D =[ 3 4 1 -6 21 3 8 1 5] D =[ 3 4 1 8 1 1 8 1 5] EB = D ED = B EB = F EF = B

First use Theorem 1.4.5 and then use the inversion algorithm to find \(A^{-1}\), if it exists. a.\(A=\left[\begin{array}{ll}1 & 4 \\2 & 7\end{array}\right]\) b. \(A=\left[\begin{array}{rr}2 & -4 \\-4 & 8\end{array}\right]\) Equation Transcription: [] [] Text Transcription: A^-1 A =[2 7 1 4] A =[-4 8 2 -4] Text Transcription:

First use Theorem 1.4.5 and then use the inversion algorithm to find \(A^{-1}\), if it exists. a. \(A=\left[\begin{array}{rr}1 & -5 \\3 & -16\end{array}\right]\) b. \(A=\left[\begin{array}{rr}6 & 4 \\-3 & -2\end{array}\right]\) Equation Transcription: [] [] Text Transcription: A^-1 A =[3 -16 1 -5] A =[-3 -2 6 4]

Use the inversion algorithm to find the inverse of the matrix (if the inverse exists). a. \(\left[\begin{array}{lll}1 & 2 & 3 \\2 & 5 & 3 \\1 & 0 & 8\end{array}\right]\) b. \(\left[\begin{array}{rrr}-1 & 3 & -4 \\2 & 4 & 1 \\-4 & 2 & -9\end{array}\right]\) Equation Transcription: [] [] Text Transcription: [1 0 8 2 5 3 1 2 3] [-4 2 -9 2 4 1 -1 3 -4]

Use the inversion algorithm to find the inverse of the matrix (if the inverse exists). a. \(\left[\begin{array}{rrr}\frac{1}{5} & \frac{1}{5} & -\frac{2}{5} \\\frac{1}{5} & \frac{1}{5} & \frac{1}{10} \\\frac{1}{5} & -\frac{4}{5} & \frac{1}{10}\end{array}\right]\) b. \(\left[\begin{array}{rrr}\frac{1}{5} & \frac{1}{5} & -\frac{2}{5} \\\frac{2}{5} & -\frac{3}{5} & -\frac{3}{10} \\\frac{1}{5} & -\frac{4}{5} & \frac{1}{10}\end{array}\right]\) Equation Transcription: [] [] Text Transcription: [15 -45 1 1015 15 110 15 15 -25] [15 -45 1 1025 -35 -310 15 15 -25]

Use the inversion algorithm to find the inverse of the matrix (if the inverse exists). \(\left[\begin{array}{lll}1 & 0 & 1 \\0 & 1 & 1 \\1 & 1 & 0\end{array}\right]\) Equation Transcription: [] Text Transcription: [1 1 0 0 1 11 0 1]

Use the inversion algorithm to find the inverse of the matrix (if the inverse exists). \(\left[\begin{array}{ccc}\sqrt{2} & 3 \sqrt{2} & 0 \\-4 \sqrt{2} & \sqrt{2} & 0 \\0 & 0 & 1\end{array}\right]\) Equation Transcription: [] Text Transcription: [ 0 0 1 -4 sqrt 2 sqrt 2 0 sqrt 2 3sqrt 2 0]

Use the inversion algorithm to find the inverse of the matrix (if the inverse exists). \(\left[\begin{array}{lll}2 & 6 & 6 \\2 & 7 & 6 \\2 & 7 & 7\end{array}\right]\)\(\) Equation Transcription: [] Text Transcription: [ 2 7 7 2 7 6 2 6 6]

Use the inversion algorithm to find the inverse of the matrix (if the inverse exists). \(\left[\begin{array}{llll}1 & 0 & 0 & 0 \\1 & 3 & 0 & 0 \\1 & 3 & 5 & 0 \\1 & 3 & 5 & 7\end{array}\right]\) Equation Transcription: [] Text Transcription: [1 3 5 71 3 5 01 3 0 01 0 0 0]

Use the inversion algorithm to find the inverse of the matrix (if the inverse exists). \(\left[\begin{array}{rrrr}2 & -4 & 0 & 0 \\1 & 2 & 12 & 0 \\0 & 0 & 2 & 0 \\0 & -1 & -4 & -5\end{array}\right]\) Equation Transcription: [] Text Transcription: [0 -1 -4 -5 0 0 2 0 1 2 12 0 2 -4 0 0]

Use the inversion algorithm to find the inverse of the matrix (if the inverse exists). \(\left[\begin{array}{rrrr}0 & 0 & 2 & 0 \\1 & 0 & 0 & 1 \\0 & -1 & 3 & 0 \\2 & 1 & 5 & -3\end{array}\right]\) Equation Transcription: [] Text Transcription: [2 1 5 -3 0 -1 3 0 1 0 0 1 0 0 2 0]

Find the inverse of each of the following \(4 × 4\) matrices, Where \(k_{1}, k_{2}, k_{3}, k_{4}\), and \(k\) are all nonzero. a. \(\left[\begin{array}{cccc}k_{1} & 0 & 0 & 0 \\0 & k_{2} & 0 & 0 \\0 & 0 & k_{3} & 0 \\0 & 0 & 0 & k_{4}\end{array}\right]\) b.. \(\left[\begin{array}{llll}k & 1 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & k & 1 \\0 & 0 & 0 & 1\end{array}\right]\) Equation Transcription: [] [] Text Transcription: 4 x 4 k_1, k_2, k_3, k_4 k [0 0 0 k_4 0 0 k_3 0 0 k_2 0 0 k_1 0 0 0 ] [0 0 0 1 0 0 k 1 0 1 0 0 k 1 0 0 ]

Find the inverse of each of the following \(4 × 4\) matrices, Where \(k_{1}, k_{2}, k_{3}, k_{4}\), and \(k\) are all nonzero. a. \(\left[\begin{array}{cccc}0 & 0 & 0 & k_{1} \\0 & 0 & k_{2} & 0 \\0 & k_{3} & 0 & 0 \\k_{4} & 0 & 0 & 0\end{array}\right]\) b. \(\left[\begin{array}{cccc}k & 0 & 0 & 0 \\1 & k & 0 & 0 \\0 & 1 & k & 0 \\0 & 0 & 1 & k\end{array}\right]\) Equation Transcription: [] [] Text Transcription: 4 x 4 k_1, k_2, k_3, k_4 k [k_4 0 0 0 0 k_3 0 0 0 0 k_2 0 0 0 0 k_1 ] [0 0 1 k 0 1 k 0 1 k 0 0 k 0 0 0 ]

Find all values of \(c\), if any, for which the given matrix is invertible. \(\left[\begin{array}{lll}c & c & c \\1 & c & c \\1 & 1 & c\end{array}\right]\) Equation Transcription: [] Text Transcription: c [ 1 1 c 1 c c c c c ]

Find all values of \(c\), if any, for which the given matrix is invertible. \(\left[\begin{array}{lll}c & 1 & 0 \\1 & c & 1 \\0 & 1 & c\end{array}\right]\) Equation Transcription: [] Text Transcription: c [ 0 1 c 1 c 1 c 1 0 ]

Express the matrix and its inverse as products of elementary matrices. \(\left[\begin{array}{rr}-3 & 1 \\2 & 2\end{array}\right]\) Equation Transcription: [] Text Transcription: [ 2 2 -3 1]

Express the matrix and its inverse as products of elementary matrices. \(\left[\begin{array}{rr}1 & 0 \\-5 & 2\end{array}\right]\) Equation Transcription: [] Text Transcription: [-5 2 1 0]

In Exercises 23–26, express the matrix and its inverse as products of elementary matrices. \(\begin{array}{ccc} 1 & 0 & -2 \\ 0 & 4 & 3 \\ 0 & 0 & 1 \end{array}\) Equation Transcription: Text Transcription: 1 0 -2 0 4 3 0 0 1

In Exercises 23–26, express the matrix and its inverse as products of elementary matrices. \(\begin{array}{lll} 1 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & 1 \end{array}\) Equation Transcription: Text Transcription: 1 1 0 1 1 1 0 1 1

In Exercises 27–28, show that the matrices ???? and ???? are row equivalent by finding a sequence of elementary row operations that produces ???? from ????, and then use that result to find a matrix ???? such that ???????? = ????. \(A=\left[\begin{array}{lll} 1 & 2 & 3 \\ 1 & 4 & 1 \\ 2 & 1 & 9 \end{array}\right], \quad B=\left[\begin{array}{ccc} 1 & 0 & 5 \\ 0 & 2 & -2 \\ 1 & 1 & 4 \end{array}\right]\) Equation Transcription: Text Transcription: A=[1 1 3 _ 1 4 1 _ 2 1 9 ], B=[1 0 5 _ 0 2 -2 _ 1 1 4 ]

In Exercises 27–28, show that the matrices ???? and ???? are row equivalent by finding a sequence of elementary row operations that produces ???? from ????, and then use that result to find a matrix ???? such that ???????? = ????. \(A=\left[\begin{array}{ccc} 2 & 1 & 0 \\ -1 & 0 & 0 \\ 3 & 0 & -1 \end{array}\right], \quad B=\left[\begin{array}{ccc} 6 & 9 & 4 \\ -5 & -1 & 0 \\ -1 & -2 & -1 \end{array}\right]\) Equation Transcription: [], [] Text Transcription: A=[ 2 1 0 _ -1 0 0 _ 3 0 -1 ], B=[ 6 9 4 _ -5 -1 0 _ -1 -2 -1 ]

Show that if \(A=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & c \end{array}\right]\) is an elementary matrix, then at least one entry in the third row must be zero. Equation Transcription: Text Transcription: A= [ 1 0 0 _ 0 1 0 _ a b c ]

Show that \(A=\left[\begin{matrix}0&a&0&0&0\\ b&0&c&0&0\\ 0&d&0&e&0\\ 0&0&f&0&g\\ 0&0&0&h&0\end{matrix}\right]\) is not invertible for any values of the entries. Equation Transcription: Text Transcription: A=[0 a 0 0 0 _ b 0 c 0 0 _ 0 d 0 e 0 _ 0 0 f 0 g _ 0 0 0 h 0 ]

Prove that if \(A\) and \(B\) are \(m \times n\) matrices, then \(A\) and \(B\) are row equivalent if and only if \(A\) and \(B\) have the same reduced row echelon form. Equation Transcription: Text Transcription: A B m times n

Prove that if \(A\) is an invertible matrix and \(B\) is row equivalent to \(A\), then \(B\) is also invertible. Equation Transcription: Text Transcription: A B

Prove that if \(B\) is obtained from \(A\) by performing a sequence of elementary row operations, then there is a second sequence of elementary row operations, which when applied to \(B\) recovers \(A\). Equation Transcription: Text Transcription: A B

In parts \((a)-(g)\) determine whether the statement is true or false, and justify your answer. a. The product of two elementary matrices of the same size must be an elementary matrix. Equation Transcription: Text Transcription: (a)-(g)

In parts \((a)-(g)\) determine whether the statement is true or false, and justify your answer. b. Every elementary matrix is invertible. Equation Transcription: Text Transcription: (a)-(g)

If \(A\) and \(B\) are row equivalent, and if \(B\) and \(C\) are row equivalent, then \(A\) and \(C\) are row equivalent. Equation Transcription: Text Transcription: A B C

If \(A\) is an \(n \times n\) matrix that is not invertible, then the linear system \(A x=0\) has infinitely many solutions. Equation Transcription: Text Transcription: A n times n Ax=0

If \(A\) is an \(n \times n\) matrix that is not invertible, then the matrix obtained by interchanging two rows of \(A\) cannot be invertible. Equation Transcription: Text Transcription: A n times n

If \(A\) is invertible and a multiple of the first row of \(A\) is added to the second row, then the resulting matrix is invertible. Equation Transcription: Text Transcription: A

An expression of an invertible matrix \(A\) as a product of elementary matrices is unique. Equation Transcription: Text Transcription: A

It can be proved that if the partitioned matrix \(\left[\begin{array}{ll} A & B \\ C & D \end{array}\right]\) is invertible, then its inverse is \(\left[\begin{array}{lc} A^{-1}+A^{-1} B\left(D-C A^{-1} B\right)^{-1} C A^{-1} & -A^{-1} B\left(D-C A^{-1} B\right)^{-1} \\ -\left(D-C A^{-1} B\right)^{-1} C A^{-1} & \left(D-C A^{-1} B\right)^{-1} \end{array}\right]\) provided that all of the inverses on the right side exist. Use this result to find the inverse of the matrix \(\left[\begin{array}{cccc} 1 & 2 & 1 & 0 \\ 0 & -1 & 0 & 1 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 3 & 3 \end{array}\right]\) Equation Transcription: Text Transcription: [A B _ C D ] [A^-1 +A^-1 B(D-CA^-1 B)^-1 CA^-1 -A^-1 B(D-CA^-1 B)^-1 _ -(D-CA^-1 B)^-1 CA^-1 (D-CA^-1 B)^-1 [1 2 1 0 _ 0 -1 0 1 _ 0 0 2 0 _ 0 0 3 3 ]