?In Exercises 23–24, find the diagonal entries of ???????? by

In Exercises 1–2, classify the matrix as upper triangular, lower triangular, or diagonal, and decide by inspection whether the matrix is invertible. Recall that a diagonal matrix is both upper and lower triangular, so there may be more than one answer in some parts. \(\text { a. }\left[\begin{array}{ll} 2 & 1 \\ 0 & 3 \end{array}\right]\) \(\text { b. }\left[\begin{array}{ll} 0 & 0 \\ 4 & 0 \end{array}\right]\) \(\text { C. }\left[\begin{array}{lll} -1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{array}\right]\) \(\text { d. }\left[\begin{array}{lll} 3 & -2 & 7 \\ 0 & 0 & 3 \\ 0 & 0 & 8 \end{array}\right]\) Equation Transcription: Text Transcription: a. [2 1 _ 0 3 ] b. [0 0 _ 4 0 ] c. [-1 0 0 _ 0 2 0 _ 0 0 1/5 ] d. [3 -2 7 _ 0 0 3 _ 0 0 8 ]

In Exercises 1–2, classify the matrix as upper triangular, lower triangular, or diagonal, and decide by inspection whether the matrix is invertible. Recall that a diagonal matrix is both upper and lower triangular, so there may be more than one answer in some parts. \(\text { a. }\left[\begin{array}{ll} 4 & 0 \\ 1 & 7 \end{array}\right]\) \(\text { b. }\left[\begin{array}{cc} 0 & -3 \\ 0 & 0 \end{array}\right]\) \(\text { C. }\left[\begin{array}{ccc} 4 & 0 & 0 \\ 0 & \frac{3}{5} & 0 \\ 0 & 0 & -2 \end{array}\right]\) \(d .\left[\begin{array}{lll} 3 & 0 & 0 \\ 3 & 1 & 0 \\ 7 & 0 & 0 \end{array}\right]\) Equation Transcription: [] [] Text Transcription: a. [4 0 _ 1 7 ] b. [0 -3 _ 0 0 ] c. [4 0 0 _ 0 3/5 0 _ 0 0 -2 ] d. [3 0 0 _ 3 1 0 _ 7 0 0 ]

In Exercises 3–6, find the product by inspection. \(\left[\begin{array}{rrr} 3 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 2 \end{array}\right]\left[\begin{array}{cc} 2 & 1 \\ -4 & 1 \\ 2 & 5 \end{array}\right]\) Equation Transcription: Text Transcription: [3 0 0 _ 0 -1 0 _ 0 0 2 ] [2 1 _ -4 1 _ 2 5 ]

In Exercises 3–6, find the product by inspection. \(\left[\begin{array}{lrr} 1 & 2 & -5 \\ -3 & -1 & 0 \end{array}\right]\left[\begin{array}{ccc} -4 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 2 \end{array}\right]\) Equation Transcription: Text Transcription: [1 2 -5 _ -3 -1 0 ] [-4 0 0 _ 0 3 0 _ 0 0 2 ]

In Exercises 3–6, find the product by inspection. \(\left[\begin{array}{ccc} 5 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -3 \end{array}\right]\left[\begin{array}{ccccc} -3 & 2 & 0 & 4 & -4 \\ 1 & -5 & 3 & 0 & 3 \\ -6 & 2 & 2 & 2 & 2 \end{array}\right]\) Equation Transcription: Text Transcription: [5 0 0 _ 0 2 0 _ 0 0 -3 ] [-3 2 0 4 -4 _ 1 -5 3 0 3 _ -6 2 2 2 2 ]

In Exercises 3–6, find the product by inspection. \(\left[\begin{array}{ccc} 2 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 4 \end{array}\right]\left[\begin{array}{crr} 4 & -1 & 3 \\ 1 & 2 & 0 \\ -5 & 1 & -2 \end{array}\right]\left[\begin{array}{rrr} -3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 2 \end{array}\right]\) Equation Transcription: Text Transcription: [2 0 0 _ 0 -1 0 _ 0 0 4 ] [4 -1 3 _ 1 2 0 _ -5 1 -2 ] [-3 0 0 _ 0 5 0 _ 0 0 2 ]

In Exercises 7–10, find \(A^{2}, A^{-2}, \text { and } A^{-k}\)(where k is any integer) by inspection. \(A=\left[\begin{array}{rr} 1 & 0 \\ 0 & -2 \end{array}\right]\) Equation Transcription: Text Transcription: A^2 , A^-2 , and A^-k A=[1 0 _ 0 -2 ]

In Exercises 7–10, find \(A^{2}, A^{-2}, \text { and } A^{-k}\)(where k is any integer) by inspection. \(A=\left[\begin{array}{rrr} -6 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{array}\right]\) Equation Transcription: Text Transcription: A^2 , A^-2 , and A^-k A=[-6 0 0 _ 0 3 0 _ 0 0 5 ]

In Exercises 7–10, find \(A^{2}, A^{-2}, \text { and } A^{-k}\)(where k is any integer) by inspection. \(A=\left[\begin{array}{ccc} \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{3} & 0 \\ 0 & 0 & \frac{1}{4} \end{array}\right]\) Equation Transcription: Text Transcription: A^2 , A^-2 , and A^-k A=[1/2 0 0 _ 0 1/3 0 _ 0 0 1/4 ]

In Exercises 7–10, find \(A^{2}, A^{-2}, \text { and } A^{-k}\) (where k is any integer) by inspection. \(A=\left[\begin{array}{cccc} -2 & 0 & 0 & 0 \\ 0 & -4 & 0 & 0 \\ 0 & 0 & -3 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right]\) Equation Transcription: Text Transcription: A^2 , A^-2 , and A^-k A=[-2 0 0 0 _ 0 -4 0 0 _ 0 0 -3 0 _ 0 0 0 2 ]

In Exercises 11–12, compute the product by inspection. \(\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 3 \end{array}\right]\left[\begin{array}{lll} 2 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 0 \end{array}\right]\left[\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{array}\right]\) Equation Transcription: Text Transcription: [1 0 0 _ 0 0 0 _ 0 0 3 ] [2 0 0 _ 0 5 0 _ 0 0 0 ] [0 0 0 _ 0 2 0 _ 0 0 1 ]

In Exercises 11–12, compute the product by inspection. \(\left[\begin{array}{lll} -1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{array}\right]\left[\begin{array}{ccc} 3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 7 \end{array}\right]\left[\begin{array}{ccc} 5 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 3 \end{array}\right]\) Equation Transcription: Text Transcription: [-1 0 0 _ 0 2 0 _ 0 0 4 ] [3 0 0 _ 0 5 0 _ 0 0 7 ] [5 0 0 _ 0 -2 0 _ 0 0 3 ]

In Exercises 13–14, compute the indicated quantity. \(\left[\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right]^{39}\) Equation Transcription: Text Transcription: [1 0 _ 0 -1 ]^39

In Exercises 13–14, compute the indicated quantity. \(\left[\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right]^{1000}\) Equation Transcription: Text Transcription: [1 0 _ 0 -1 ]^1000

In Exercises 15–16, use what you have learned in this section about multiplying by diagonal matrices to compute the product by inspection. \(\text { a. }\left[\begin{array}{ccc} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{array}\right]\left[\begin{array}{cc} u & { }^{v} \\ w & x \\ y & z \end{array}\right] \text { b. }\left[\begin{array}{ccc} r & s & t \\ u & y & w \\ x & y & z \end{array}\right]\left[\begin{array}{ccc} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{array}\right]\) Equation Transcription: Text Transcription: a. [a 0 0 _ 0 b 0 _ 0 0 c ] [u v _ w x _ y z ] b. [r s t _ u v w _ x y z ] [a 0 0 _ 0 b 0 _ 0 0 c ]

In Exercises 15–16, use what you have learned in this section about multiplying by diagonal matrices to compute the product by inspection. \(\text { a. }\left[\begin{array}{cc} u & y \\ w & x \\ y & z \end{array}\right]\left[\begin{array}{cc} a & 0 \\ 0 & b \end{array}\right] \quad \text { b. }\left[\begin{array}{lll} a & 0 & 0 \\ 0 & b & 0 & 0 \end{array}\right]\left[\begin{array}{ccc} r & s & t \\ u & y & w \\ x & y & z \end{array}\right]\) Equation Transcription: Text Transcription: a. [u v _ w x _ y z ] [a 0 _ 0 b ] b. [a 0 0 _ 0 b 0 _ 0 0 c ] [r s t _ u v w _ x y z ]

In Exercises 17–18, create a symmetric matrix by substituting appropriate numbers for the ×’s. \(\text { a. }\left[\begin{array}{rr} 2 & -1 \\ x & 3 \end{array}\right]\) \(\text { b. }\left[\begin{array}{cccc} 1 & x & x & x \\ 3 & 1 & x & x \\ 7 & -8 & 0 & x \\ 2 & -3 & 9 & 0 \end{array}\right]\) Equation Transcription: Text Transcription: a. [2 -1 _ x 3 ] b. [1 x x x _ 3 1 x x _ 7 -8 0 x _ 2 -3 9 0 ]

In Exercises 17–18, create a symmetric matrix by substituting appropriate numbers for the ×’s. \(\text { a. }\left[\begin{array}{ll} 0 & x \\ 3 & 0 \end{array}\right]\) \(\text { b. }\left[\begin{array}{rrrr} 1 & 7 & -3 & 2 \\ x & 4 & 5 & -7 \\ x & x & 1 & -6 \\ x & x & x & x \end{array}\right]\) Equation Transcription: Text Transcription: a. [0 x _ 3 0 ] b. [1 7 -3 2 _ x 4 5 -7 _ x x 1 -6 _ x x x x ]

In Exercises 19–22, determine by inspection whether the matrix is invertible. \(\left[\begin{array}{lll} 0 & 6 & -1 \\ 0 & 7 & -4 \\ 0 & 0 & -2 \end{array}\right]\) Equation Transcription: Text Transcription: [0 6 -1 _ 0 7 -4 _ 0 0 -2 ]

In Exercises 19–22, determine by inspection whether the matrix is invertible. \(\left[\begin{array}{rrr} -1 & 2 & 4 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{array}\right]\) Equation Transcription: Text Transcription: [-1 2 4 _ 0 3 0 _ 0 0 5 ]

In Exercises 19–22, determine by inspection whether the matrix is invertible. \(\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 2 & -5 & 0 & 0 \\ 4 & -3 & 4 & 0 \\ 1 & -2 & 1 & 3 \end{array}\right]\) Equation Transcription: Text Transcription: [1 0 0 0 _ 2 -5 0 0 _ 4 -3 4 0 _ 1 -2 1 3 ]

In Exercises 19–22, determine by inspection whether the matrix is invertible. \(\left[\begin{array}{cccc} 2 & 0 & 0 & 0 \\ -3 & -1 & 0 & 0 \\ -4 & -6 & 0 & 0 \\ 0 & 3 & 8 & -5 \end{array}\right]\) Equation Transcription: Text Transcription: [2 0 0 0 _ -3 -1 0 0 _ -4 -6 0 0 _ 0 3 8 -5 ]

In Exercises 23–24, find the diagonal entries of ???????? by inspection. \(A=\left[\begin{array}{ccc} 3 & 2 & 6 \\ 0 & 1 & -2 \\ 0 & 0 & -1 \end{array}\right], \quad B=\left[\begin{array}{ccc} -1 & 2 & 7 \\ 0 & 5 & 3 \\ 0 & 0 & 6 \end{array}\right]\) Equation Transcription: Text Transcription: A=[3 2 6 _ 0 1 -2 _ 0 0 -1 ], B=[-1 2 7 _ 0 5 3 _ 0 0 6 ]

In Exercises 23–24, find the diagonal entries of ???????? by inspection. \(A=\left[\begin{array}{ccc} 4 & 0 & 0 \\ -2 & 0 & 0 \\ -3 & 0 & 7 \end{array}\right], B=\left[\begin{array}{ccc} 6 & 0 & 0 \\ 1 & 5 & 0 \\ 3 & 2 & 6 \end{array}\right]\) Equation Transcription: Text Transcription: A=[4 0 0 _ -2 0 0 _ -3 0 7 ], B=[6 0 0 _ 1 5 0 _ 3 2 6 ]

In Exercises 25–26, find all values of the unknown constant(s) for which ???? is symmetric. \(A=\left[\begin{array}{lr} 4 & -3 \\ a+5 & -1 \end{array}\right]\) Equation Transcription: Text Transcription: A=[4 -3 _ a+5 -1 ]

In Exercises 25–26, find all values of the unknown constant(s) for which ???? is symmetric. \(A=\left[\begin{array}{ccc} 2 & a-2 b+2 c & 2 a+b+c \\ 3 & 5 & a+c \\ 0 & -2 & 7 \end{array}\right]\) Equation Transcription: Text Transcription: A=[2 a-2b+2c 2a+b+c _ 3 5 a+c _ 0 -2 7 ]

In Exercises 27–28, find all values of x for which ???? is invertible. \(A=\left[\begin{array}{ccc} x-1 & x^{2} & x^{4} \\ 0 & x+2 & x^{3} \\ 0 & 0 & x-4 \end{array}\right]\) Equation Transcription: Text Transcription: A=[x-1 x^2 x^4 _ 0 x+2 x^3 _ 0 0 x-4 ]

In Exercises 27–28, find all values of x for which ???? is invertible. \(A=\left[\begin{array}{ccc} x-\frac{1}{2} & 0 & 0 \\ x & x-\frac{1}{3} & 0 \\ x^{2} & x^{3} & x+\frac{1}{4} \end{array}\right]\) Equation Transcription: Text Transcription: A=[x-1/2 0 0 _ x x-1/3 0 _ x^2 x^3 x+1/4 ]

If ???? is an invertible upper triangular or lower triangular matrix, what can you say about the diagonal entries of \(A^{-1}\) ? Equation Transcription: Text Transcription: A^-1

Show that if ???? is a symmetric \(n \times n\) matrix and ???? is any \(n \times m\) matrix, then the following products are symmetric: \(B^{T} B, B B^{T}, B^{T} A B\) Equation Transcription: Text Transcription: n times n n times m B^T B, BB^T , B^T AB

In Exercises 31–32, find a diagonal matrix \(A\) that satisfies the given condition. \(A^{5}=\left[\begin{array}{rrr}1 & 0 & 0 \\0 & -1 & 0 \\0 & 0 & -1\end{array}\right]\) Equation Transcription: [? Text Transcription: A A5=[ _0 0 -1 ^0 -1 0 1^ 0 0?

In Exercises 31–32, find a diagonal matrix \(A\) that satisfies the given condition. \(A^{-2}=\left[\begin{array}{lll}9 & 0 & 0 \\0 & 4 & 0 \\0 & 0 & 1\end{array}\right]\) Equation Transcription: [? Text Transcription: A A^-2=[ _0 0 1 ^0 4 0 ^9 0 0?

Verify Theorem 1.7.1(b) for the matrix product \(AB\) and Theorem 1.7.1(d) for the matrix \(A\) where \(A=\left[\begin{array}{rrr}-1 & 2 & 5 \\0 & 1 & 3 \\0 & 0 & -4\end{array}\right], \quadB=\left[\begin{array}{rrr}2 & -8 & 0 \\0 & 2 & 1 \\0 & 0 & 3\end{array}\right]\) Equation Transcription: [], [] Text Transcription: A AB A=[ _0 0 -4 ^0 1 3 ^-1 2 5], B=[ _0 0 3 ^0 2 1 ^2 -8 0]

Let \(A\) be an \(n \times n\) symmetric matrix. a. Show that \(A^{2}\) is symmetric. b. Show that \(2 A^{2}-3 A+I\) is symmetric. Equation Transcription: Text Transcription: A n x n A^2 2A^2-3A+I

Verify Theorem 1.7.4 for the given matrix \(A\). a. \(A=\left[\begin{array}{rr}2 & -1 \\-1 & 3\end{array}\right]\) b. \(A=\left[\begin{array}{rrr}1 & -2 & 3 \\-2 & 1 & -7 \\3 & -7 & 4\end{array}\right]\) Equation Transcription: [] [] Text Transcription: A A=[_-1 3 ^ 2 -1] A=[ _3 -7 4 ^-2 ^1 -7 ^1 -2 3]

Find all \(3 \times 3\) diagonal matrices \(A\) that satisfy \(A^{2}-3 A-4 I=0\) Equation Transcription: Text Transcription: A 3 x 3 A^2-3A-4I=0

Let \(A=\left[a_{i j}\right]\) be an \(n \times n\) matrix. Determine whether \(A \)is symmetric. a. \(a_{i j}=i^{2}+j^{2}\) b. \(a_{i j}=i^{2}-j^{2}\) c. \(a_{i j}=2 i+2 j\) d. \(a_{i j}=2 i^{2}+2 j^{3}\) Equation Transcription: Text Transcription: A=[a_ij] A nn a_ij=i^2+j^2 a_ij=i^2-j^2 a_ij=2i+2j a_ij=2i^2+2j^3

On the basis of your experience with Exercise 37, devise a general test that can be applied to a formula for \(a_{i j}\) to determine whether \(A=\left[a_{i j}\right]\) is symmetric. Equation Transcription: Text Transcription: a_ij A=[a_ij]

Find an upper triangular matrix that satisfies \(A^{3}=\left[\begin{array}{rr}1 & 30 \\0 & -8\end{array}\right]\) Equation Transcription: [] Text Transcription: A3=[_0 -8 ^1 30]

If the \(n \times n\) matrix \(A\) can be expressed as \(A=L U\), where \(L\) is a lower triangular matrix and \(U\) is an upper triangular matrix, then the linear system \(A \mathbf{x}=\mathbf{b}\) can be expressed as \(L U \mathbf{x}=\mathbf{b}\) and can be solved in two steps: Step 1. Let \(U \mathbf{x}=\mathbf{y}\), so that \(L U \mathbf{x}=\mathbf{b}\) can be expressed as \(L \mathbf{y}=\mathbf{b}\). Solve this system. Step 2. Solve the system \(U \mathbf{x}=\mathbf{y} \text { for } \mathbf{x}\). In each part, use this two-step method to solve the given system. a. \(\left[\begin{array}{rrr}1 & 0 & 0 \\-2 & 3 & 0 \\2 & 4 & 1\end{array}\right]\left[\begin{array}{rrr}2 &-1 & 3 \\0 & 1 & 2 \\0 & 0 & 4\end{array}\right]\left[\begin{array}{l}x_{1} \\x_{2} \\x_{3}\end{array}\right]=\left[\begin{array}{r}1 \\-2 \\0\end{array}\right]\) b. \(\left[\begin{array}{rrr}2 & 0 & 0 \\4 & 1 & 0 \\-3 & -2 & 3\end{array}\right]\left[\begin{array}{rrr}3 &-5 & 2 \\0 & 4 & 1 \\0 & 0 & 2\end{array}\right]\left[\begin{array}{l}x_{1} \\x_{2} \\x_{3}\end{array}\right]=\left[\begin{array}{r}4 \\-5 \\2\end{array}\right]\) Equation Transcription: ???? = ???? ???? = ???? ???? = ???? ???? = ???? ????= ???? ???? = ???? for ???? [] [] []= [] [] [] []= [] Text Transcription: n x n A A=LU L U Ax = b LUx = b Ux = y LUx = b Ly= b Ux = y for x [ _2 4 1 ^-2 3 0 ^1 0 0] [ _0 0 4 ^0 1 2 ^2-1 3] [ _x_3 ^x_2 ^x_1]= [ _0 ^-2 ^1] [ _-3 -2 3 ^4 1 0 ^2 0 0] [ _0 0 2 ^0 4 1 ^3 -5 2] [_ x_3 ^x_2 ^x_1]= [ _ 2 ^-5 ^4]

In the text we defined a matrix \(A\) to be symmetric if \(A^{T}=A\). Analogously, a matrix \(A\) is said to be skew-symmetric if \(A^{T}=-A\). Exercises 41–45 are concerned with matrices of this type. Fill in the missing entries (marked with \(\times\) ) so the matrix \(A\) is skew-symmetric. a. \(A=\left[\begin{array}{rrr}\times & \times & 4 \\0 & \times & \times \\\times & -1 & \times\end{array}\right]\) b. \(A=\left[\begin{array}{rrr}\times & 0 & \times \\\times & \times & -4 \\8 & \times & \times\end{array}\right]\) Equation Transcription: [] [] Text Transcription: A A^T=A A A^T=-A A A=[ _x -1 x ^0 x x ^x x 4] A=[ _8 x x ^x x -4 ^x 0 x ]

In the text we defined a matrix \(A\) to be symmetric if \(A^{T}=A\). Analogously, a matrix \(A\) is said to be skew-symmetric if \(A^{T}=-A\). Exercises 41–45 are concerned with matrices of this type. Find all values of a, b, c, and d for which ???? is skew-symmetric. \(A=\left[\begin{array}{rcc}0 & 2 a-3 b+c & 3 a-5 b+5 c \\-2 & 0 & 5 a-8 b+6 c \\-3 & -5 & d\end{array}\right]\) Equation Transcription: and [] Text Transcription: A A^T=A A A^T=-A a,b,c, and d A A=[_-3 -5 d ^-2 0 5a-8b+6c ^0 2a-3b+c 3a-5b+5c]

In the text we defined a matrix \(A\) to be symmetric if \(A^{T}=A\). Analogously, a matrix \(A\) is said to be skew-symmetric if \(A^{T}=-A\). Exercises 41–45 are concerned with matrices of this type. We showed in the text that the product of symmetric matrices is symmetric if and only if the matrices commute. Is the product of commuting skew-symmetric matrices skew-symmetric? Explain. Equation Transcription: Text Transcription: A A^T=A A A^T=-A

In the text we defined a matrix \(A\) to be symmetric if \(A^{T}=A\). Analogously, a matrix \(A\) is said to be skew-symmetric if \(A^{T}=-A\). Exercises 41–45 are concerned with matrices of this type. Prove that every square matrix A can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix. [Hint: Note the identity \(A=\frac{1}{2}\left(A+A^{t}\right) \frac{1}{2}\left(A-A^{t}\right)\).] Equation Transcription: Text Transcription: A A^T=A A A^T=-A A A=1/2(A+A^t)1/2(A-A^t)

In the text we defined a matrix \(A\) to be symmetric if \(A^{T}=A\). Analogously, a matrix \(A\) is said to be skew-symmetric if \(A^{T}=-A\). Exercises 41–45 are concerned with matrices of this type. Prove the following facts about skew-symmetric matrices. a. If \(A\) is an invertible skew-symmetric matrix, then \(A^{-1}\) is skew-symmetric. b. If \(A\) and \(B\) are skew-symmetric matrices, then so are \(A^{T}, A+B, A-B\), and \(kA\) for any scalar \(k\). Equation Transcription: Text Transcription: A A^T=A A A^T=-A A A^-1 A B A^T, A+B, A-B kA k

Prove: If the matrices \(A\) and \(B\) are both upper triangular or both lower triangular, then the diagonal entries of both \(AB\) and \(BA\) are the products of the diagonal entries of \(A\) and \(B\). Equation Transcription: Text Transcription: A B AB BA A B

Prove: If \(A^{T} A=A\), then \(A\) is symmetric and \(A=A^{2}\) . Equation Transcription: Text Transcription: A^T A=A A A=A^2

In parts (a)–(m) determine whether the statement is true or false, and justify your answer. a. The transpose of a diagonal matrix is a diagonal matrix.

In parts (a)–(m) determine whether the statement is true or false, and justify your answer. b. The transpose of an upper triangular matrix is an upper triangular matrix.

In parts (a)–(m) determine whether the statement is true or false, and justify your answer. c. The sum of an upper triangular matrix and a lower triangular matrix is a diagonal matrix.

In parts (a)–(m) determine whether the statement is true or false, and justify your answer. d. All entries of a symmetric matrix are determined by the entries occurring on and above the main diagonal.

In parts (a)–(m) determine whether the statement is true or false, and justify your answer. e. All entries of an upper triangular matrix are determined by the entries occurring on and above the main diagonal.

In parts (a)–(m) determine whether the statement is true or false, and justify your answer. f. The inverse of an invertible lower triangular matrix is an upper triangular matrix.

In parts (a)–(m) determine whether the statement is true or false, and justify your answer. g. A diagonal matrix is invertible if and only if all of its diagonal entries are positive.

In parts (a)–(m) determine whether the statement is true or false, and justify your answer. h. The sum of a diagonal matrix and a lower triangular matrix is a lower triangular matrix.

In parts (a)–(m) determine whether the statement is true or false, and justify your answer. i. A matrix that is both symmetric and upper triangular must be a diagonal matrix.

In parts (a)–(m) determine whether the statement is true or false, and justify your answer. j. If \(A\) and \(B\) are \(n \times n\) matrices such that \(A+B\) is symmetric, then \(A\) and \(B\) are symmetric. Equation Transcription: Text Transcription: A B n x n A+B A B

In parts (a)–(m) determine whether the statement is true or false, and justify your answer. k. If \(A\) and \(B\) are \(n \times n\) matrices such that \(A+B\) is upper triangular, then \(A\) and \(B\) are upper triangular. Equation Transcription: Text Transcription: A B n x n A+B A B

In parts (a)–(m) determine whether the statement is true or false, and justify your answer. l. If \(A^{2}\) is a symmetric matrix, then \(A\) is a symmetric matrix. Equation Transcription: Text Transcription: A^2 A

In parts (a)–(m) determine whether the statement is true or false, and justify your answer. m. If \(k A\) is a symmetric matrix for some \(k \neq 0\), then \(A\) is a symmetric matrix. Equation Transcription: Text Transcription: kA k neq 0 A

Starting with the formula stated in Exercise T1 of Section 1.5, derive a formula for the inverse of the “block diagonal” matrix \(\left[\begin{array}{cc}D_{1} & 0 \\0 & D_{2}\end{array}\right]\) in which \(D_{1}\) and \(D_{2}\) are invertible, and use your result to compute the inverse of the matrix \(M=\left[\begin{array}{cccc}1.24 & 2.37 & 0 & 0 \\3.08 & -1.01 & 0 & 0 \\0 & 0 & 2.76 & 4.92 \\0 & 0 & 3.23 & 5.54\end{array}\right]\) Equation Transcription: [] [] Text Transcription: [_0 D_2 ^D_1 0] D_1 D_2 [ _0 0 3.23 5.54 ^0 0 2.76 4.92 ^3.08 -1.01 0 0 ^1.24 2.37 0 0]