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

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

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Q: Find a (2x2) singular matrix A that maps (2, 4) into (3, 1).

A: Singular matrix mean its Determinant is equal to 'zero' .

Q: must be a non-invertible matrix.

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Q: Compute the product 0 () 0 (²) as a 2 x 2 matrix. What kind of isometry is this product?

A: Given product is  Oπ5O2π15 where Oθ represents rotation matrix through an angle θ in anti-clockwise…

Q: Using only 0's and l's, list all possible 2 × 2 matrices in row canonical fom.

A: Row canonical matrix All zero rows, if any, are at the bottom of the matrix. Each leading non zero…

Q: Which one of the matrices below is singular( Not invertible )?

A: A matrix whose determinant is zero is called as a singular matrix. Let us verify the options. (a)…

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

A: .

Q: The 2 ´ 2 identity matrix is:

A: To find:

Q: matrix with eig satisfies

A:

Q: Is it possible for A3 to be an identity matrix without A beinginvertible? Explain.

A: To explain:It is possible for A3 to be an identity matrix without A being invertible.By definition

Q: The i th row vector of a matrix product AB can be computed by multiplying A by the ith row vector of…

A: Given statement is  The i th row vector of a matrix product AB can be computed by multiplying A by…

Q: Under which conditions an inverse matrix be defined for a matrix?

A: There are two conditions for matrix to be invertible.

Q: Prove that if an upper triangular matrix is orthogonal, then it must be a diagonal matrix

A: Solution: If the multiplication of matrix with its transpose is equal to the identity, then a 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: Prove Theorem For any matrix A, AAT and ATA are symmetric matrices.

A: To Prove for any Matrix A, AAT and ATA are symmetric matrices.   A matrix X is symmteric if XT=X.…

Q: What is the multiplicative identity matrix?

A: According to the question, we have to define the multiplicative identity matrix. As the above…

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 any matrix of size 3 by 3, can be written as the sum of two invertible matrices, that is,…

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Q: * Find the Matrix B from order 2 x 4 , Its elements are written in the form by = i + ()

A: Here we find matrix B of order 2×4

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

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Q: Is there any 2x2 matrix with all non-zero entries, that is not diagonalizable but invertible? If…

A: Solution

Q: Show that the matrix below does not have an LU-factorization.

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

A: Given matrix combination is B = A + I

Q: What does it mean to say that A is a matrix with dimensionm  n?

A: A matrix is a rectangular array of elements arranged in rows and columns. The size of the matrix is…

Q: Let A be mxh matrix. an Prove that. AlL %3D max

A: (Solving the one question as per our guidelines) Solution: Given A is an m×n matrix. We have to…

Q: verify that the matrices and scalars in Exer- cise 1 satisfy the stated properties. (a) (AT)" = A…

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Q: Find the rank and nullity theorem for the following matrix(in image).

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Q: Show that the diagonal entries of a Hermitian matrix must be real.

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Q: Define what it means for a function Q : R" R to be a quadratic form. Furthermore, explain how we…

A: The quadratic form is one of the most important parts of linear algebra. Quadratic means degree two.…

Q: Let A = [0 1 0 0 ], A^(-1) is equal to: * Does not exist Identity matrix Null matrix

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Q: Let X and Y be symmetric matrices both size 15x15. Decide if Z = XT(X+Y) is a symmetric matrix too.…

A: Given, X & Y are symmetric matrices. So, XT = X and, YT = Y   Given, Z = XT(X+Y) So, ZT =…

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: find if the vector x is in the range for matrix A

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Q: write down in se quence steps in finding the the inverse OF matrix using the adjoint the matrix. OF

A: Let's find.

Q: What is the condition for a diagonal matrix to be orthogonal?

A: use the properties of orhtogonal with respect to a diagonal matrix...

Q: ers such that p2 e matrix

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Q: Explain transpose AT of the m x n matrix A?

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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: Show that a triangular matrix is normal if it is diagonal

A: Given: A triangular matrix is normal.if it is diagonal

Q: 2. for the augmanted matrix

A:

Q: Find a 2 × 1 matrix x with entries not all zero such that

A: Given: Ax=3x, where A=2112 let's assume matrix x is x=ab

Q: Let A be an m xn matrix. Explain why the matrix ATA is symmetric.

A: A square matrix is symmetric if  A^T = A

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: Find the change of basis matrix from B2 to B1. B2

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Q: Construct a nonzero 2 × 2 matrix that is invertible but not diagonalizable.

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Q: Show that the product of two orthogonal matrices is orthogonal.

A: Note:  We are allowed to solve one question at a time so kindly upload rest of the questions as next…

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: The i-th column vector of an AB matrix product can be calculated by multiplying A by the i-th B…

A: True Because take A=a11a12a21a22 and B=b11b12b21b22 Then the product of AB is,…

Q: Determine whether the matrix is invertible. 5 5 -5 8 2 -8 -7 0 7 Also find AT

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Q: Since it is, find the matrix P ^ -1 by using the adjoint matrix of the P matrix.

A: We are given that  We need to find the inverse of p-matrix