Suppose that {v1, v2,..., vn} is an orthonormal set in Rn with the standard inner product. Let the matrix A be given by A = v1 v2 ··· vn . Show that A is non- singular and compute its inverse. Give three different ex- amples of such a matrix in R2 or R3.

Q: Prove that for an n×n matrix A, if A^2 is invertible, then A is invertible.

A: Since, if a matrix A is non-zero then A is invertible.

Q: Prove that if one row (column) of the n × n matrix A consists entirely of zeros, then A is singular.

A: We know that if all entries of one row(column) of the n×n is zero then determinant of matrix is…

Q: Let A be any m×n matrix, Omn be the m×n zero matrix, and c be scalar. Show that if cA = Omn then…

A: Let A be matrix of order m× n Then A = aij   where i is row and j is column ,   i = 1, 2, . . . m…

Q: Suppose A and B are n×n matrices such that B is invertible and AB is invertible. Prove that A is…

A: Given that the matrices A and B are of order n×n and both A, AB are invertible.

Q: Let A be an m × n matrix. Show that if x ∈ N(ATA), then Ax is in both R(A) and N(AT ).

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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 be an n × n real matrix. Prove that if 01,.., ơn are the singular values of A, then of, ..., o…

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Q: Show that if A is a symmetric matrix, then AT is symmetric.

A: we have to show that if A is a symmetric matrix, then AT is symmetric.   as we know that a matrix B…

Q: Show that if A is any mxn matrix, then Im= A.

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Q: Prove that if A is an m x n matrix of rank r, then the dimension of the solution space of Ax = 0 is…

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Q: Prove that if A is an invertible n × n matrix, then the rows of A span R“ by using transpose.

A: we use the fact, if A is invertible then A is row equivalent to the identity matrix I.

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

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Q: Prove that if an n × n matrix A is not invertible, then A[adj(A)] is the zero matrix.

A: 1). Matrix A is invertible if and only if matrix A is a non singular matrix. That is, determinant of…

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: Given that A, B, C are all square matrices of the same size, matrices A and C are both orthonormal,…

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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: .

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Q: Let A be an n ×n matrix. Prove that dim(span({In,A,A2, . . .})) ≤n.

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Q: Let A be a nonsingular n×n matrix with a nonzero cofactor Ann, and set c = det(A)/Ann Show that if…

A: Let A is any non singular nxn matrix with non zeroCofactor Ann and Set C = det(A)Ann   By concept…

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: Let A=[aij ] be an m xn matrix and k be a scalar then kA is equal to : * [kaij]mxn [aij/k]mxn…

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Q: Let P = A(ATA)−1AT, where A is an m × n matrixof rank n. Show that P is symmetric.

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Q: Let A and B be n × n matrices such that AB is singular. Prove that either A or B is singular.

A: Let A and B be n × n matrices such that AB is singular.

Q: Let A be an n × n singular matrix. Then

A: if matrix A is singular that means its determined is zero.so at least one row of matrix A is…

Q: Let A be an m × n matrix. Prove that if AT A is nonsingu- lar, then rank A = n.

A: Given that, A is an n x n matrix. A^T is the transpose matrix of A. AT A is non-singular. We have to…

Q: Let A be an orthogonal matrix. Show that cA is orthogo- nal if and only if c = ±1.

A: Given:            A be an orthogonal matrix.

Q: Let A ∈ Mn × n (R) such that A3 = In × n and A≠In × n Give an example of a 3 × 3 matrix that…

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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 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…

A:

Q: Let A and B be n x n matrices such that AB is invertible. Prove that A and B are invertible. Give an…

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Q: Refer to the following theorem: If A is an mxn matrix with linearly independent columns, then A can…

A: Solution

Q: (b) By replacing A with another matrix B different from A in only a single entry, define an inner…

A: To prove a function to be an inner product, we have to show it satisfies the following five…

Q: Let A be an n × n matrix with a column of zeros. Show directly from the Leibniz formula that…

A: Let, A be an n×n matrix with a column of zeros.  To prove detA=0 using Leibnitz formula for…

Q: Show that if A is an upper triangular matrix, then AT is lower triangular.

A: Given, A is an upper triangular matrix

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 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: If B is invertible matrix the show det B = det B that %3D

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Q: Construct a 4 x 3 matrix A = [ āj az āz ] such that {ā1, ā2 } is a linearly independent set and az…

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Q: Explain why for any matrix A, AA' exists and prove that AA' is a symmetric matrix

A: Let A is any matrix of order m by n Am×n    and   Atn×m Now  AAt exists of order m by m To prove…

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 A be an m × n matrix. Show that ATA and AAT are both symmetric.

A: Given A be an m×n matrix. we have to show that ATA and AAT are both symmetric.we will make use of…

Q: Suppose that A e 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.

Q: Let A be a symmetric matrix and P be an orthogonal matrix. Show that A is positive semi-definite if…

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Q: Let A, be the n x n matrix each of whose entries is 1. Show that if n > 1, then In - An is…

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Q: Suppose A is nxn mabrix Cequre mabria) prove that if the pivobe for the matrix is Unique ornoto…

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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: show that if At exists for matrix A > then A is to

A:

Q: Prove that if L is an invertible lower triangular matrix, then L-l is also lower triangular.

A: handwriting solution