If the n x n matrices A and B are orthogonal, which of the matrices in Exercises 5 through 11 must be orthogo- nal as well?

Q: Construct a 3 x 3 triangular matrix A so that 1] the vector 2 is in Col A. 3]

A: We know every lower or upper traingular matrix is a triangular matrix.

Q: Show that if the entries in each row of an n × n matrix A add up to zero, then the column vectors of…

A: linearly dependent means det of A is zero

Q: the n x n matrices A and B are orthogonal, which of e matrices in Exercises 5 through 11 must be…

A:

Q: Let C and D be n x n skew-symmetric matrices. Prove that if CD = – -DC, then CD is a skew-symmetric…

A: Given problem:- Let C and D be n x n skew-symmetric matrices. Prove that if CD=-DC, then CD is a…

Q: and = %3D are oignualus of a matrix A Corraspondeng to the egnualuong=2 and >=-2 then = ACTa t Ja)=…

A: For a given matrix A if 'x' is eigen vector of matrix A corresponding to the eigen value ' q ' then…

Q: if A is (m*n) matrix, prove 'rank(A)=n-dim(N(A))

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Q: Under what conditions will the diagonal matrix be invertible? Assume that A is invertible and find…

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Q: Given the partitioned matrix A = [ B]. B]₁ C D in which B and C are invertible. Which of the…

A:

Q: Find matrices P and D so that A = PDP1 and D is a diagonal

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Q: Let A be an n × n matrix and let Q and V be n × northogonal matrices. Show that ||AV ||2 = || A ||2

A:

Q: Let A be an n × n matrix and let Q and V be n × northogonal matrices. Show that ||QA ||2 = ||A ||2

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Q: Let X Give an example of two 2 × 2 matrices A and B, neither of which is the identy matrix I, such…

A: Ok

Q: Find a 2 x 2 matrix B + 0 and B + 1, such that AB = BA, where A How many such matri- 2 ces B are…

A: Given data: The matrix is given by "A". A=1221 Consider the matrix "B". B=abcd According to the…

Q: Express the following matrix as a product of several elemen- tary matrices. 1 3 E 2 3 ||

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Q: Suppose that A and B are matrices having n rows. Prove that M(A|B) = (MA|MB) for any m ×n matrix M.

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Q: Suppose B = QRQ-1, where Q is orthogonal, and R is upper triangular. Show that if B is symmetric,…

A: Given that B=QRQ-1 where, Q is orthogonal and R is upper triangular. Then, multiply Q on both sides:…

Q: If A is an orthogonal matrix, find a QR factorization of A

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Q: Find diagonal matrix D is similar with A a A =[? -5 And make sure can transtor diagomalizablewith

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Q: If the n x n matrices A and B are symmetric and B is in- vertible, which of the matrices in…

A: (16) Given:- A and B are symmetric  ⇒AT=A,    BT=B B is invertible ⇒B-1 exists To prove :- A+B is…

Q: Let D = P'AP be a diagonal matrix, wher %3D Pesell thet if D:

A: Given, D=P-1AP be a diagonal matrix, where P is an orthogonal matrix. To prove that A is symmetric.

Q: If the rows of an n X n matrix A are linearly inde­pendent as vectors in Rn, what is the rank of A?…

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Q: Show that if A is an orthogonal matrix, then det(A) = ±1.

A: Given If A is an orthogonal matrix. To show det(A) = ±1

Q: Suppose that A € Mnxn(R) is a positive-definite symmetric matrix. Since A is symmetric, A is or-…

A: We will use the knowledge of standard value decomposition (SVD) of a square matrix.

Q: If A is a 6 × 8 matrix, what is the smallest possible dimension of Nul A?

A: Consider the provided matrix, Given, A is a 6 x 8 matrix. Since, the rank of A equals the number of…

Q: Let A be an n × n matrix in which the entries of each row sum to zero. Find det(A).

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Q: Find a 3 × 2 matrix T, such that TT' = A, where 4 A = |4 3 3 5

A: Given: To find a 3 x 2 matrix T, such that TT' = A,  A = 540453035

Q: Let A and B be matrices with size 3 x 3. If only one of these matrices are diagonalizable and if…

A: Given, A and B be matrices with size 3×3det(A-λI3)=(λ-2)(λ2-9)det(B-λI3)=(λ-3)2(λ+2)(a) Eigenvalues…

Q: Use the SVD to show that any square matrix A can be written as A = unitary and P is Hermitian. UP…

A: By SVD, we know that matrix A can be written as A=SDVTWhere S and V are orthogonal matrices andD is…

Q: Assuming that all matrices aren xnand invertible, solve for D. BCA DCB'A = BC?.

A: Properties of invertible matrix:  If a matrix A is invertible and then there must exist a matrix…

Q: If a matrix A E Mn(R) and rank(A) = n, is it true that the column vectors of A are linearly…

A:

Q: Determine whether the given matrix is orthogonal. If it is, find its inverse. in given fig.

A: Determine whether the given matrix is orthogonal. If it is, find its inverse. Let…

Q: Let A be an m×n matrix, X an n×r matrix, and B an m × r matrix. Show that AX = B if and only if Axj…

A:

Q: Show that if A is an orthogonal matrix, then det(A) = +1.

A: To prove, detA=±1

Q: Determine the dimensions of Nul A and Col A for the matrices. A =

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Q: Let A = [aij] be a square matrix If B is obtained by adding a multiple of one row (column) of A to…

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Q: Let A be an invertible, n x n matrix such that A² = A. a) Calculate det(A). b) If n = 3, what is A?…

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Q: Suppose A = UDVT where U and V are n x n matrices with the property that UTU = I and %3D vTV = I,…

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Q: If the n x n matrices A and B are symmetric and B is in- vertible, which of the matrices in…

A: Q(14).

Q: Let A and B square matrices where detA=4 and detB=12. What is det(A-1B)?

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Q: Determine the matrix C such that (CT – 21) 1 1 2 2 3

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Q: If A is a matrix in Rmxn with {()} -5 C(A) = span 2 %3D -2 what is m?

A: If A is a matrix in R^(m×n) then A is a linear transformation form n to m

Q: If the n x n matrices A and B are symmetric and B is in- vertible, which of the matrices in…

A:

Q: Find all a and B >0 so that the matrix B = 23 6 2 is diagonally dominant;

A: A square matrix is diagonally dominant if   for .

Q: Let A = [a;;] be the n x n matrix defined by aji = k and a;; = 0 if i + j. Show that if B is any n x…

A: Given: A=aijn×n,     aii=k and aij=0 if i≠k. Let B be a square matrix of order 3 which can be…

Q: If U is a square matrix such that UTU I, what can we say about det (U)?

A:

Q: The mxn zero matrix , denoted by 0mn or simply 0 , is the matrix whose elements are all zero , find…

A: A matrix is a mathematical object containing numbers arranged in rows and columns. The matrix in…

Q: Construct a nonzero 3 × 3 matrix A and a nonzero vector b such that b is in Col A, but b is not the…

A: We will construct A and put b as a linear combination of column vectors in A.