Let A and B be m × n matrices. (a) Suppose that v"Aw = v"Bw for all vectors v, w with real entries. Prove that %3D

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A: linearly dependent means det of A is zero

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A: Given that the n×n non-singular matrices as A and B. It is known that a matrix is non-singular only…

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Q: the matrIX A into a product QDQ" where Q is orthogonal and D is diagonal.

A: Given,          A=-5-1-1-1-3-1-1-1-3

Q: Let A be square matrix. If A" = 0. A= Oava %3D for n> 2. Then

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Q: Suppose that Q is an orthogonal matrix, then O a lQ°| = 4 O b. Q°| = 2 O d. Q°| = 0

A: This question is related to Linear Algebra

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Q: "Let A, B be n x n matrices. If 3A(B“) is invertible, then both A and B are invertible."

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Q: Let A be an m×n matrix. Explain why the followingare true. Any vector b ∈ Rm can be uniquely written…

A: For  matrix A, N(A) is the null space of A. R(A) is the column space of the matrix A and R(AT) is…

Q: Let x and y be distinct vectors in Rn (i.e., x = y), and let A be an n × n matrix with the property…

A:

Q: Is it true or false? If A is nxn matrix, c is scalar, then cA |=c | A| Birini seçin:

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Q: Find three non-trivial 2 x 2 matrices such that the square of the matrix is the zero matrix, i.e: A?

A: We want to find matrix A such that A2 = 0000

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Q: Let A be an (n x n) matrix and za column vector. If AA = I then Ax|| = ||r||. %3D %3D O True O False

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Q: Consider A and B are two square matrices and B is a scalar. Calculate (BA^T B)^(-1) = *

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Q: Let A and B be square matrices of order n satisfying Ax = Bx for all x in Rn. (a) Find the rank and…

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Q: Suppose A is (\m \times n \) matrix that has r pivots, then the null space of A contains only the…

A: Option A will be the right answer.

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Q: Show that the column vectors of following matrix A are linearly independent:

A: the given question is taken from the matrix in which we have to find show that the given matrix A is…

Q: Show that if A is a mxn matrix and rank(A)=m then Ax=b has a solution for any vector b in Fm (Hint:…

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Q: (a) Let A, B be n × n symmetric matrices such that XTAX = X™BX for all column vectors X e R". Show…

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Q: 6. Find the distance between the matrices A and B, p(A, B), using the ||- ||2 for the matrix -[:] -1…

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Q: If A and B are square matrices of the same order, then (AB)T = AT BT True False

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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: Let the matrices Ho, H, Ha 1x1 M어ni x [1]. be defined ... the He=L He-I vector of length n-2" can be…

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Q: Г1 3 2 5 1 [3 4 2

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Q: Show that if the non-zero vectors s(1), s(2),..., s(m) are conjugate to a positive definite matrix…

A: We are given that the non-zero vectors s(1), s(2), ..., s(n) are conjugate to a positive definite…

Q: If A is an m × n matrix of rank r, what are the dimensions of N(A) and N(AT )? Explain.

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

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Q: Find conditions on w, x, y, and z such that AB = BA for the matrices below.

A: Here we find the condition w,x,y,z such that AB=BA where A=wxyz , B=11-11

Q: If xER" is a column vector (i.e. an n ×1 matrix), then In-2xx' is always a symmetric matrix. True…

A: Consider the given column matrix as x=x1x2⋮⋮xn-1xn, x∈Rn. Compute the transpose of the matrix x as…

Q: Let A and B be matrices of size p × m and n × q , respectively. Assume that mn = pq . Define R: Mmn…

A: Matrix A is p×m and B is n×q In a m×n matrix number of element will be m.n R is defined as,…

Q: Let A be an n x n Hermitian matrix and X and Y be n x 1 column vectors. Prove that (AX)·Y – X ·(AY)=…

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Q: Let A = [kaij]nxn, B = [aij]nxn, be an nxn matrices and k be a scalar then det(A) is equal to: *…

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Q: 5. Let x and y be vectors in R" and let Q be an n x n orthogonal matrix. Show that if z = Qx and w =…

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Q: If a vector is in the range of a matrix A, then it is also in the null space of A. true of false?

A: Answer

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A: Given: All the given matrices are n × n. A and B are row equivalent matrices.

Q: Find 2x2 invertible matrices A and B such that A + B not equal to 0 and A + B is not invertible.

A: Wee need to find matrix A and B and do their addition and find invertible

Q: If A is an n×n matrix and {v1,v2,…,vn}    is a linearly independent set of vectors in Rn, then…

A: This is true

Q: Let the entries of a matrix A = \aij be defined as aij 2i2 – i+j+ g(j), where g is a %3D function of…

A: Given Ais a symmetric matrix therefore aij=aji…

Q: let  A and  B be  square  matrices of order n  such that  Ax=Bx  and x belongs to Rn  then   find…

A: Ax = Bx implies  Ax - Bx = O So, (A - B)x = O where x belongs to Rn  So, x belongs to the null space…

Q: Let A and B be square matrices of order 2 such thatA=-7 and B=4.What is (ABY"| %3D %3D

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Q: Show that if the non-zero vectors s1), s(2),..., s) are conjugate to a positive definite matrix G,…

A:

Q: Let A, Band C be square matrices Such that AB = AC, If A#0,then B-C. 1s this True or false? Explain…

A: Let A, B, and C be square matrices such that AB=AC. If A≠0 , then B=C. Determine true or false.

Q: 20. Can every vector in R* be written as a linear combination of the columns of the matrix B above?…

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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.

Q: let A be an mxn matrix and v and w vectors in Rn. If v and w were both solutions to Ax=0, show that…

A:

Q: Suppose A, B, and C are three non-zero square matrices such that AC = BC. Is it true that A = B? If…

A: Not true

Q: If the matrices A and B are conformable for sum A+B the product AB then prove that (A+B)" = A" +B"…

A: Prove that. (a).A+BT=AT+BT(b).ABT=BTAT