-) Let A, B be nx n symmetric matrices such that XTAX = X™BX for all %3D column vectors X e R". Show that A == B.

Q: Let A, B & Mn (C) be Hermitian matrices. Then

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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: Prove Any n by n triangular matrix with main diagonal entriesare all 1 is diagonalizable if and only…

A: Triangular Matrix: A triangular matrix is a square matrix where all the values above or below the…

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

A: Given that A and B  are n×n non singular matrices. An n×n matrix A is non singular only if A is…

Q: ) Consider the matrix A = [a b] with a, b, c and d real-valued numbers. Under which condition(s) for…

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Q: VýB × VA

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Q: Prove If U1 and U2 are both unit upper triangular matrices, then the product U1U2 is also a unit…

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Q: If A is an q x n matrix with orthonormal columns & W be the column space, then AA"y Projwy for all y…

A: Given that A is a q×n matrix with orthonormal columns. Hence there are q rows and n columns in A.…

Q: 6. Show that if Q is an orthogonal matrix, then 1 0 %3D 0 Q is also an orthogonal matrix.

A: We have to show that HHt = I

Q: Find two different 2 × 2 matrices A such that

A: Given:               A2=1001

Q: [1] Construct a 3 x 3 nonzero matrix A such that the vector |2 is a solution of 3. Ах 3D0

A: A vector is given let x=123 To find: A non-zero 3×3× matrix,A=a11a12a13a21a22a23a31a32a33

Q: 4. Prove that if A is a square matrix, then det(A™ A) = det(AA") %3D

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Q: Prove that an m X n matrix A has rank 1 if and only if A can be written as the outer product uvT of…

A: Given that an m X n matrix A has rank 1 if and only if A can be written as the outer product uvT of…

Q: Show that if A is an orthogonal matrix, then A−1 is orthogonal.

A: Explanation: Given that,  A is an orthogonal matrix.

Q: Determine the Null Space of the (3x3) matrix A for which (2j + i aij if i # j if i = j %3D i+ti

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Q: Prove that if A is an invertible matrix and B is row equiva- to A, then B is also invertible. ation

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Q: Suppose that x and y are mxn matrices. Prove that xT- yT=(X -YT. Find a 2x2matrix A, that is not the…

A: Take A and B as arbitrary m×n matrices and evaluate the left and right sided expression

Q: Let A = [aij] given by a x bij = (i-j)^3 is a: * Anti-Symmetric matrix Symmetric matrix Identity…

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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: 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: 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: 2. If A is an m xn matrix, show that every vector in null(A) is orthogonal to every vector in…

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Q: If A is an q x n matrix with orthonormal columns & W be the column space, then AA" y = Projwy for…

A: If A is an invertible qxn matrix with orthonormal columns & W be the column space, then the…

Q: Let B be an invertible matrix. Show that A = BTB is positive definite.

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Q: Prove the Frobenius theorem in the case where A is a 2 ×2 matrix.

A: This theorem is not valid for order less than 3

Q: If an nxn matrix is invertible, then the column vectors form a basis for R

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Q: If {u1, u2, U3, ... set of vectors in R" and the matrix U has columns u1, ..., Uk, then U'U is the…

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Q: Construct the matrix with rank one that has Av = 12u for v = ½(1, 1, 1, 1) and u= ½(2, 2, 1). Its…

A: Given: Let A be a given matrix. rank (A) = 1 To find: The only one singular value for  Av = 12 u .

Q: Compute the rank and nullity of the given matrices over the indicated Zp

A: Given Data The matrix is   Transform the matrix by using the row echelon form,

Q: 13. Find all 2 x 2 matrices A = (" ta² = (C :) such that %3D

A: We need to find the set of all 2×2 matrices such that A2=O, where O is the zero matrix of order 2.

Q: Let A be a 7 × 5 matrix of rank 4. Let P and Q be theprojection matrices that project vectors in R7…

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Q: Prove det(A) = det(A'), for an upper triangular matrix A.

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Q: Find a (2x2) singular matrix A that maps (s1, ss) into (s2, s6).

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Q: 2 Find the cosine of the angle between these two matrices, using the standard matrix inner product…

A: Solution

Q: If A and B are orthogonal nxn matrices, then AB-1 is an orthogonal matrix. True False

A: here we must know that  1)  AA-1=A-1A =I2) ATT=A3) AI =IA =A

Q: Let r1, .. r,be vectors in R", and let Q be an m xn matrix. Write the matrix Qr, Qr, as a product of…

A: Given:Let r1.......rp be vectors in ℝnLet Q be an m×n matrix Qr1.......QrpR=r1.......rpQ=Q…

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: .. Let A = [aij] be a 3 x 3 square matrix such that aij = 1 – |i – j|- (a) Construct the matrix A…

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Q: Let A € RPXm have rank r. Show that there are matrices C and B such that A = CB, where C e RPxr and…

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Q: 7. Show that the matrix A=I– 2vv' , where v is a unit vector in R", and I is an nxn identity matrix,…

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Q: Assume A and B are two m × n matrices with the same reduced echelon form. Show that there exists an…

A: Given A and B are two m×n matrices with the same reduced echelon form. We have to show that there…

Q: Let x and y be square matrices of order n a. Show that if x and y are skew symmetric, then x + y is…

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Q: Let A be a lower triangular n x n matrix with nonzero entries on the diagonal. Show that A is…

A: Consider A be a lower triangular matrix with non-zero entries on the diagonal.

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: Suppose that {v1, v2,..., vn} is a linearly independent set of vectors in Rn. Show that if A is an n…

A: Let's suppose we have a linear relation:…

Q: Let A be a 4x4 singular matrix. What is the smallest possible dimension of Null Space of A? A) 2

A: Using,  Rank- Nullity Thereom

Q: Let A be an m x n matrix. Prove that if B can be obtained from A by anelementary row [column]…

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Q: 2. Find the matrices P and D for an orthogonal diagonalization A=PDPT for 3 0 0 A=

A: First find eigenvalues eigenvactors corresponding to eigenvalues then we get matrices P and D.

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: Let A be a 7 × 5 matrix of rank 4. Let P and Q be theprojection matrices that project vectors in R7…

A: Let A be a 7 × 5 matrix of rank 4. Let P and Q be the projection matrices that project vectors in…