?Use the inverse of the coefficient matrix to solve the systems in 22 and 23\(4 x+3 y-2

In each of Problems 1-4, verify by multiplication that the given pair of matrices are inverses of one another. \(\left[\begin{array}{ll} 5 & 3 \\ 2 & 1 \end{array}\right]\) \(\left[\begin{array}{cc} -1 & 3 \\ 2 & -5 \end{array}\right]\) Equation Transcription Text Transcription [5 3_2 1] [-1 3_2 -5]

In each of Problems 1-4, verify by multiplication that the given pair of matrices are inverses of one another. \(\left[\begin{array}{ll} 2 & -4 \\ 2 & 0 \end{array}\right]\) \(\left[\begin{array}{ll} 0 & 1 / 2 \\ -1 / 4 & 1 / 4 \end{array}\right]\) Equation Transcription Text Transcription [2 -4_2 0] [0 1/2_-¼ ¼]

In each of Problems 1-4, verify by multiplication that the given pair of matrices are inverses of one another. \(\left[\begin{array}{ccc} 1 & 0 & 1 \\ 1 & 1 & -2 \\ 0 & 1 & 1 \end{array}\right]\) \(\left[\begin{array}{ccc} 3 / 4 & 1 / 4 & -1 / 4 \\ -1 / 4 & 1 / 4 & 3 / 4 \\ 1 / 4 & -1 / 4 & 1 / 4 \end{array}\right]\) Equation Transcription Text Transcription [1 0 1_1 1 -2_0 1 1] [¾ ¼ -1/4_-¼ ¼ 3/4_1/4 -¼ -¼]

In each of Problems 1-4, verify by multiplication that the given pair of matrices are inverses of one another. \(\left[\begin{array}{llrl} -28 & -13 & 3 \\ 2 & 1 & 0 & \\ -7 & -3 & 1 \end{array}\right]\) \(\left[\begin{array}{ccc} 1 & 4 & -3 \\ -2 & -7 & 6 \\ 1 & 7 & -2 \end{array}\right]\) Equation Transcription Text Transcription [-28 -13 3_2 1 0_-7 -3 1] [1 4 -3_-2 -7 6_1 7 -2]

Use row reduction to calculate the inverse of each matrix in problems 5-14. Consider k0 \(\left[\begin{array}{ll} 2 & 0 \\ 1 & 1 \end{array}\right]\) Equation Transcription Text Transcription [2 0_1 1]

Use row reduction to calculate the inverse of each matrix in problems 5-14. Consider k0 \(\left[\begin{array}{ll} 1 & 3 \\ 2 & 5 \end{array}\right]\) Equation Transcription Text Transcription [1 3_2 5]

Use row reduction to calculate the inverse of each matrix in problems 5-14. Consider k0 \(\left[\begin{array}{ccc} 0 & 1 & 1 \\ 5 & 1 & -1 \\ 3 & -3 & -3 \end{array}\right]\) Equation Transcription Text Transcription [0 1 1_5 1 -1_3 -3 -3]

Use row reduction to calculate the inverse of each matrix in problems 5-14. Consider k0 \(\left[\begin{array}{lll} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]\) Equation Transcription Text Transcription [0 0 1_0 1 0_1 0 0]

Use row reduction to calculate the inverse of each matrix in problems 5-14. Consider k0 \(\left[\begin{array}{lll} k & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\) Equation Transcription Text Transcription [k 0 0_0 1 0_0 0 1]

Use row reduction to calculate the inverse of each matrix in problems 5-14. Consider k0 \(\left[\begin{array}{ccc} 1 & 0 & 1 \\ 1 & -1 & 0 \\ 0 & 2 & 1 \end{array}\right]\) Equation Transcription Text Transcription [1 0 1_1 -1 0_0 2 1]

Use row reduction to calculate the inverse of each matrix in problems 5-14. Consider k0 \(\left[\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 1 & k & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\) Equation Transcription Text Transcription [1 0 0 0_0 1 k 0_0 0 1 0_0 0 0 1]

Use row reduction to calculate the inverse of each matrix in problems 5-14. Consider k0 \(\left[\begin{array}{llll} 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 2 \end{array}\right]\) Equation Transcription Text Transcription [1 0 1 1_0 0 1 0_1 1 1 0_1 0 0 2]

Use row reduction to calculate the inverse of each matrix in problems 5-14. Consider k0 \(\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 1 & -2 & 0 \\ 1 & -1 & 3 & 3 \end{array}\right]\) Equation Transcription Text Transcription [1 0 0 0_0 -1 0 0_0 1 -2 0_1 -1 3 3]

Use row reduction to calculate the inverse of each matrix in problems 5-14. Consider k0 \(\left[\begin{array}{llll} 0 & 1 & 2 & 1 \\ 4 & 0 & 1 & 2 \\ 0 & 1 & 0 & 0 \\ 0 & 2 & 0 & 1 \end{array}\right]\) Equation Transcription Text Transcription [0 1 2 1_4 0 1 2_0 1 0 0_0 2 0 1]

Verify that the inverse of a square matrix A of order two is given by the following handy formula, if \(a d-b c \neq 0\), That is, show that \(A A^{-1}=I\) \(\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]^{-1}=\frac{1}{a d-b c}\left[\begin{array}{cc} d & -b \\ -c & a \end{array}\right]\) Equation Transcription Text Transcription [a b_c d]^-1=(1/ad-bc)[d -b_-c a] Ad-bc is not equal to 0 AA^-1=I

Compute the inverse matrix \(A=\left[\begin{array}{ll} 1 & 3 \\ 1 & 2 \end{array}\right]\) By setting \(A^{-1}=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\) And solving a system of equations for a, b, c, and d. Equation Transcription Text Transcription A=[1 3_1 2] A^-1=[a b_c d]

For problems 17 and 18, answer the given question by finding a proof or a counterexample in \(M_{22}\). Is it true that the sum of invertible n x n matrices is invertible? Equation Transcription Text Transcription M_22

For problems 17 and 18, answer the given question by finding a proof or a counterexample in M22. Is it true that the only nxn matrices for which A^2=A are the zero matrix and the identity matrix? \(A^{2}=A\) Equation Transcription Text Transcription A^2=A

Show that if a matrix has an \(\text { inverse }\) it is unique. Equation Transcription Text Transcription inverse

Use the inverse matrix found in Problem 6 to solve the system \(x_{1}+3 x_{2}=-4\) \(2 x_{1}+5 x_{2}=10\) Equation Transcription Text Transcription x_1+3x_2=-4 2x_1+5x_2=10

Use the inverse matrix determined in Problem 7 to solve the system \(y+z=5\) \(5 x+y-z=2\) \(3 x-3 y-3 z=0\) Equation Transcription Text Transcription y+z=5 5x+y-z=2 3x-3y-3z=0

Use the inverse of the coefficient matrix to solve the systems in Problems 22 and 23 \(x-y+z=4\) \(x+y=1\) \(x+2 y-z=0\) Equation Transcription Text Transcription x-y+z=4 x+y=1 x+2y-z=0

Use the inverse of the coefficient matrix to solve the systems in Problems 22 and 23 \(4 x+3 y-2 z=0\) \(5 x+6 y+3 z=10\) \(3 x+5 y+2 z=2\) Equation Transcription Text Transcription 4x+3y-2z=0 5x+6y+3z=10 3x+5y+2z=2

Prove that for \(A=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\) A is not invertible if \(\mathrm{ad}=\mathrm{bc}\) Equation Transcription Text Transcription A=[a b_c d] ad=bc

In Problems 25-28, assume that A and B are invertible matrices of the same order. Simplify \(\left(A B^{-1}\right)^{-1}\) Equation Transcription Text Transcription (AB^-1)^-1

In Problems 25-28, assume that A and B are invertible matrices of the same order. Simplify \(B\left(A^{2} B^{2}\right)^{-1}\) Equation Transcription Text Transcription B(A^2B^2)^-1

In Problems 25-28, assume that A and B are invertible matrices of the same order. If \(A(B A)^{-1} \bar{x}=\bar{b}\), solve for \(\bar{x}\) Equation Transcription Text Transcription A(BA)^-1x^bar=b^bar x^bar

In Problems 25-28, assume that A and B are invertible matrices of the same order. Simplify \(\left(A^{-1}\left(B A^{-1}\right)^{-1} B A^{-1}\right)^{-1}\) Equation Transcription Text Transcription (A^-1(BA^-1)^-1BA^-1)^-1

What condition is required in order to solve for \(\bar{x}\) when \((A+B) \bar{x}=\bar{b}\) Equation Transcription Text Transcription x^right arrow (A+B)x^right arrow=b^right arrow

Prove using matrix algebra that if A, B, and C are matrices such that \(A B=A C\), and A is invertible, then \(B=C\). Equation Transcription Text Transcription AB=AC B=C

Prove that if A is invertible and B is another square matrix such that \(A B=I\) then \(B=A^{-1}\) Equation Transcription Text Transcription AB=I B=A^-1

Choose a constant k so that the matrices given in Problems 32 and 33 are invertible. If no such k exists, say so and explain your reasoning. \(\left[\begin{array}{lll} 1 & 0 & k \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\) Equation Transcription Text Transcription [1 0 k_0 1 0_0 0 1]

Choose a constant k so that the matrices given in Problems 32 and 33 are invertible. If no such k exists, say so and explain your reasoning. \(\left[\begin{array}{lll} 1 & 0 & k \\ 0 & 1 & 0 \\ k & 0 & 1 \end{array}\right]\) Equation Transcription Text Transcription [1 0 k_0 1 0_k 0 1]

Let A and B be nxn matrices (a)Show that if \(B A=I_{n}\), than \(A B=I_{n}\) (b)Show that if AB is invertible, then A must be invertible. Equation Transcription Text Transcription BA=I_n AB=I_n

Show that a diagonal matrix A is invertible if and only if all diagonal elements are nonzero. Give the form of \(A^{-1}\). Equation Transcription Text Transcription A^-1

Show that an upper triangular matrix is \(\text { invertible }\) if and only if all diagonal elements are nonzero. Equation Transcription Text Transcription invertible

If the matrix-vector equation \(A \bar{x}=\bar{b}\) is inconsistent for some \(\bar{b}\) in \(R^{n}\), what can you determine about matrix A? Equation Transcription Text Transcription Ax^right arrow=b^right arrow b^right arrow R^n

Prove the following property (stated in this section): “If A is invertible, then so is \(A^{-1}\), and \(\left(A^{-1}\right)^{-1}=A\)” Equation Transcription Text Transcription A^-1 (A^-1)^-1=A

Prove the following property (stated in this section): “If A is invertible, then so is \(A^{T}\) and \(\left(A^{T}\right)^{-1}=\left(A^{-1}\right)^{T}\).” Equation Transcription Text Transcription A^T (A^T)^-1=(A^-1)^T.”

If we perform a single row operation on an identity matrix, we obtain an elementary matrix \(E_{i n t}\), \(E_{R e p l}\) or \(E_{\text {Scale }}\). Find the elementary matrices for each of the following row operations on \(I_{3}\). (a)Interchange rows 1 and 2 (\(E_{i n t}\)) (b)Add k times row 1 to row 3 (\(E_{R e p l}\)) (c)Multiply k times row 2 (\(E_{\text {Scale }}\)) Equation Transcription Text Transcription I_3 E_int E_Repl E_Scale

Explain why all elementary matrices must be invertible. Demonstrate this property by finding the inverses of \(E_{i n t}\), \(E_{R e p l}\) and \(E_{\text {Scale }}\) in Problem 40 Equation Transcription Text Transcription E_int E_Repl E_Scale

A matrix B is defined to be similar to matrix A(denoted by B~A) if there is an invertible matrix P such that \(B=P^{-1} A P\) Matrix B is similar to itself; that is \(B \sim B\) Equation Transcription Text Transcription B~B B=(P^-1)AP

A matrix B is defined to be similar to matrix A(denoted by B~A) if there is an invertible matrix P such that \(B=P^{-1} A P\) If \(B \sim A\) then \(A \sim B\) Equation Transcription Text Transcription B~A A~B B=(P^-1)AP

A matrix B is defined to be similar to matrix A(denoted by B~A) if there is an invertible matrix P such that \(B=P^{-1} A P\) If \(A \sim B\) and\(B \sim C\), then \(\mathrm{A} \sim \mathrm{C}\) Equation Transcription Text Transcription A~B B~C A~C B=(P^-1)AP

A matrix B is defined to be similar to matrix A(denoted by B~A) if there is an invertible matrix P such that \(B=P^{-1} A P\) If \(B=P^{-1} A P\) for some invertible P, Then \(B^{n}=P^{-1} A^{n} P\) for any positive integer n. Equation Transcription Text Transcription B=(P^-1)AP B^n=(P^-1)(A^n)P

If true, explain. If false, give a 2x2 counter example (a)A diagonal matrix is invertible if and only if its diagonal elements are nonzero. True or False? (b)An upper triangular matrix is invertible if and only if its diagonal elements are nonzero. True or False? (c)If A and B are nxn matrices such that A is invertible, then \(A B A^{-1}=B\). True or False? Equation Transcription Text Transcription ABA^-1=B

Find the total outputs for each input-output matrix in Problems 47-50 with demands as given \(T=\left[\begin{array}{lr}0.5 & 0 \\0 & 0.5\end{array}\right]\) \(\bar{d}=\left[\begin{array}{l}10 \\10\end{array}\right]\) Equation Transcription Text Transcription T=[0.5 0_0 0.5] =[10_10]

Find the total outputs for each input-output matrix in Problems 47-50 with demands as given \(T=\left[\begin{array}{cc}0 & 0.1 \\0.2 & 0\end{array}\right]\) \(\bar{d}=\left[\begin{array}{l}10 \\10\end{array}\right]\) Equation Transcription Text Transcription T=[0 0.1_0.2 0] =[10_10]

Find the total outputs for each input-output matrix in Problems 47-50 with demands as given \(T=\left[\begin{array}{ll}0.2 & 0.5 \\0.5 & 0.2\end{array}\right]\) \(\bar{d}=\left[\begin{array}{l}10 \\10\end{array}\right]\) Equation Transcription Text Transcription T=[0.2 0.5_0.5 0.2] =[10_10]

Find the total outputs for each input-output matrix in Problems 47-50 with demands as given \(T=\left[\begin{array}{ll}0.5 & 0.2 \\0.1 & 0.3\end{array}\right]\) \(\bar{d}=\left[\begin{array}{l}50 \\50\end{array}\right]\) Equation Transcription Text Transcription T=[0.5 0.2_0.1 0.3] =[50_50]

In Example 5, suppose that A produces $150 worth of its product and B produces $250 worth of its product. What is the dollar \(\text { value }\) of both products available for external consumption? Equation Transcription Text Transcription value

In 1966, Leontief used his input-output model to analyze the israeli economy by dividing it into three segments: agriculture(A), manufacturing(M), and energy(E), as shown in the following technological matrix. The export demands on the Israeli economy (in thousands of Israeli pounds) are listed as follows 1. Find \(I-T\) where T is the technological matrix 2. Use computer software to find \((I-T)^{-1}\) 3. Find the total output for each sector required to meet both internal and external demand Equation Transcription Text Transcription I-T (I-T)^-1

Suggested Journal Entry Discuss the similarities and dissimilarities between matrices and real numbers. Consider such issues as \(\text { invertibility }\) and cancellation laws. Can you think of comparisons going beyond the features so far presented in the text? To what extent can this comparison be adapted to the complex numbers instead? Equation Transcription Text Transcription invertibility