Let MS be the n × n matrix whose j th column is v j and let MT be the n × n matrix whose j th column is w j . Prove that MS and MT are nonsingular.

Q: Find two 3x3 invertible matrices A and B, such that A+B is not invertible.

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Q: Let Q be an orthogonal matrix such that QA makes sense. Show that A and QA have the same singular…

A: Let,

Q: Let A be an nxn matrix and B= A+ I, where / is the identity matrix, then it is possible for A and B…

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Q: Let Q be an m x n matrix with columns forming an orthonormal set. Show that (Qx) · (Qy) = x · y for…

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Q: Let A, B be 2 x 2 matrices, C a 3 x 3 matrix and D a 1 × 1 matrix. Find the condition on these…

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Q: Let A = [aj be a 4 x 4 matrix. Develop the general expression for det(A).

A: Given:               Matrix A is : [aij] is the 4×4 matrix;   General expression for the determinant…

Q: Let A be a lower triangular matrix with elements aij. Then aij = 0 when O izj O isj O i> j O i<j

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Q: Let A and B be n × n nonsingular matrices. Find nonsingular matrices P and Q so that PAQ = B .

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

Q: Let the matrix below be Anti-symr

A: Anti symmetric matrix is also known as skew symmetric matrix.

Q: 32. Construct a nondiagonal 2 x 2 matrix that is diagonalizable but not invertible.

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Q: Let A and B be n × n matrices and let x be a vector in Rn. How many scalar additions and…

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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: Give an example of a 2 x 2 matrix A such that im(A) =ker(A). %3D

A: Null space of a linear transformation: T:V(F)→V'(F) be a linear transformation then the null space…

Q: Let Q be a matrix with orthonormal columns, then QQT = 1. :Select one True False

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Q: Find an LU-factorization of the matrix A such that A = LU. %3D 6 4 -2 1 -2 3 2. -2

A: Explanation of the answer is as follows

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Q: Let be the skew operator that transforms a 3x1 column matrix into the 3x3 skew- symmetric matrix…

A: Let us consider  3×1 matrix  n=xyz and we have given [ ]x be the skew symmetric matrix…

Q: Construct a 3 x3 nonzero matrix A such that the vector is a solution of Ax = 0. 1 LA =

A: x=-1-1  1 Now let A=abcdefghi be the required vector. Then, Ax=0⇒abcdefghi-1-1  1=000⇒-a-b+c-d-e+f…

Q: * Let A be and m x n matrix. Describe the matrix E such that EA is A with row swapped.

A: Given matrix A is a m×n matrix. And the elementary matrix E and EA are row swapped. Since elementary…

Q: Show that if annxn matrix U satisfies (Ux)· (Uy) = x•y for all x and y in Rn, then U is an…

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Q: Show that if C is a matrix whose columns are the components (x1, y1) and (x2, y2) of two…

A: Concept: A branch of mathematics which deals with symbols and the rules for manipulating those…

Q: Let A be an n × n matrix and let x and y be vectors in Rn. Show that if Ax = Ay and x /= y, then the…

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Q: Let A be an Hermitian matrix and let B = iA. Show that B is skew Hermitian.

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Q: Find the standard matrix for the linear operator T: R^3 --> R^3 that first rotates a vector…

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Q: Find all aclh for which the matrix A = [at3 2 b non-singular.

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Q: Let A be a m xn matrix with columns c1, ... , en. Prove that {c1, ..., cn} is independent if and…

A: To prove  c1,c2,...cn is independent if and only if null A=0

Q: Let A be an m × n matrix such that ATA is invertible. Show that the columns of A are linearly…

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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: Let A be an m × n matrix of rank n and let P =A(ATA)−1AT . If b ∈ R(A)⊥, show that Pb = 0.

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Q: Let Q be an m xn matrix with n >n and orthonormal columns, then we have both QTQ = I, and QQT = Im…

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Q: Explain why the columns of an n×n matrix A span R^n when A is invertible.

A: Consider, a matrix A of order n, which is a invertible matrix.Since, the set of vectors

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 = UEVT be a SVD of a square matrix A. What is the product of the diagonal elements of E, in…

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Q: Let  A be a symmetric tridiagonal matrix (i.e., A is symmetric and aij = 0 when ever |i−j| > 1). Let…

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Q: Let A and B be n x n matrices.  Suppose that AB = On where On is a zero matrix of size n x n, and A…

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Q: Find Q and R in the matrix factorization 3 2 4 1 -1 QR. %3D [0 0 As usual, Q is an orthogonal matrix…

A: Given

Q: If A is a square matrix whose rows add up to the zero vector, explain why A cannot be invertible.

A: Given A is square matrix whos rows add up to zero vector.

Q: Find a (2x2) singular matrix A that maps (2, 2) into (1, 2).

A: Here in this question find a 2×2 matrix that maps (2,2)  into (1,2). Here i attached solution of…

Q: TRUE/FALSE Assume that all the given matrices are n × n. If A and B are row equivalent matrices,…

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

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: Let A be an n x n matrix and let D be a n x n diagonal matrix. Show that AD =…

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Q: Let A and B be n × n matrices and assume that columns of B are linearly dependent. Show that columns…

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Q: Let A be annxn matrix and B= A+T, where I is the identity matrix, then it is possible for A and B to…

A: Let λ be the eigen value of A. As B=A+I So, eigen value of B will be λ+1 which is not same as λ

Q: Let A and B be n x n matrices and c be a scalar. Prove that, if B is obtained by multiplying a row…

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Q: Let A be a nilpotent matrix of order n then A*n = 0 ) nxA = 0 A =nx1, l is Identity matrix

A: A be a nilpotent matrix of order n

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 be an n × n matrix and let B = A + AT and C = A − AT (a) Show that B is symmetric and C is…

A: given: let A be a n×n matrix. B=A+AT C=A-AT we have to show that B is symmetric matrix and C is skew…

Q: If A is a skew-symmetric matrix then show that AAT = AT A and A² is symmetric. [Note AT denotes the…

A: We will use definition

Q: Let Q be an orthogonal matrix such that QA makes sense.                          Show that (QA)+ =…

A: Given, A be a matrix and Q be an orthogonal matrix such that QA makes sense. We have to use product…