For matrices, a mulitplicative identity is a square matrix X such XA = AX = A...

For matrices, a mulitplicative identity is a square matrix X such XA = AX = A for any square matrix A. Prove that X must be the identity matrix.

Prove that for any invertible matrix A, the inverse matrix must be unique. Hint: Assume that there are two inverses and then show that they much in fact be the same matrix.

Prove Theorem which shows that Gauss-Jordan Elimination produces the inverse matrix for any invertible matrix A. Your proof cannot use elementary matrices (like the book’s proof does).

Prove that null(A) is a vector space.

Prove that col(A) is a vector space.

Expert Solution

Colby Messinger answered 2 years ago

for the system equation of x' = Ax if Coefficients Matrix A be ? = [...

for the system equation of x' = Ax if Coefficients Matrix A be ? = [ 5 −5 −5 −1 4 2 3 −5 −3 ] , find the basic matrix

(a) The n × n matrices A, B, C, and X satisfy the equation AX(B +...

(a) The n × n matrices A, B, C, and X satisfy the equation AX(B + CX) ?1 = C Write an expression for the matrix X in terms of A, B, and C. You may assume invertibility of any matrix when necessary. (b) Suppose D is a 3 × 5 matrix, E is a 5 × c matrix, and F is a 4 × d matrix. Find the values of c and d for which the statement “det(DEF) =...

build a program which performs matrix multiplication on square matrices. use UNIX "time" to capture the...

build a program which performs matrix multiplication on square matrices. use UNIX "time" to capture the time it takes to run the program with different data sizes. Languages: Python Task: Create matrix multiplication Input: Size of square matrix. Size should be 250, 500, 1000, 1500, 2000 Internals: Explicitly or implicitly allocate sufficient memory to hold three NxN floating point Matrices, using a random number generator -- populate two of the Matrices, Multiply the two matrices, putting the result into the...

Divide and Conquer (Strassen’s Matrix Multiplication) Given two square matrices A and B of size n...

Divide and Conquer (Strassen’s Matrix Multiplication) Given two square matrices A and B of size n x n each, find their multiplication matrix. Naive Method Following is a simple way to multiply two matrices. void multiply(int A[][N], int B[][N], int C[][N]) { for (int i = 0; i < N; i++) { for (int j = 0; j < N; j++) { C[i][j] = 0; for (int k = 0; k < N; k++) { C[i][j] += A[i][k]*B[k][j]; }...

4. The product y = Ax of an m n matrix A times a vector x...

4. The product y = Ax of an m n matrix A times a vector x = (x1; x2; : : : ; xn)T can be computed row-wise as y = [A(1,:)*x; A(2,:)*x; ... ;A(m,:)*x]; that is y(1) = A(1,:)*x y(2) = A(2,:)*x ... y(m) = A(m,:)*x Write a function M-file that takes as input a matrix A and a vector x, and as output gives the product y = Ax by row, as denoted above (Hint: use a for...

Let A be an m x n matrix. Prove that Ax = b has at least...

Let A be an m x n matrix. Prove that Ax = b has at least one solution for any b if and only if A has linearly independent rows. Let V be a vector space with dimension 3, and let V = span(u, v, w). Prove that u, v, w are linearly independent (in other words, you are being asked to show that u, v, w form a basis for V)

2. Using matrices, create an algorithm that takes a matrix of dimension N x N and...

2. Using matrices, create an algorithm that takes a matrix of dimension N x N and feed it in a spiral shape with the sequential number from 1 to N^2. Then do an algorithm in PSEint

Suppose A is an n × n matrix with the property that the equation Ax =...

Suppose A is an n × n matrix with the property that the equation Ax = b has at least one solution for each b in R n . Explain why each equation Ax = b has in fact exactly one solution

Assingment: for c++ A magic square is an n x n matrix in which each of...

Assingment: for c++ A magic square is an n x n matrix in which each of the integers 1, 2, 3...n2 appears exactly once and all column sums, row sums, and diagonal sums are equal. For example, the attached table shows the values for a 5 x 5 magic square in which all the rows, columns, and diagonals add up to 65. The following is a procedure for constructing an n x n magic square for any odd integer n....

Suppose the system AX = B is consistent and A is a 6x3 matrix. Suppose the...

Suppose the system AX = B is consistent and A is a 6x3 matrix. Suppose the maximum number of linearly independent rows in A is 3. Discuss: Is the solution of the system unique?

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