Let A be an n × n symmetric matrix of rank r. Explain why the spectral decomposition of A represents A as the sum of r rank 1 matrices.Let A be an n × n symmetric matrix of rank r. Explain why the spectral decomposition of A represents A as the sum of r rank 1 matrices.

Q: Find the general form of the span of th e indicated matrices, span(A1, A2)

A: We have to find the general form of the span of the indicated matrices, span(A1, A2)

Q: Find the spectral decomposition of the matrix 可

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Q: Let E and F be n × n elementary matrices and let C = EF. Is C nonsingular? Explain.

A: Let E and F are elementary matrices and let

Q: If A is an invertible matrix that is orthogonally diago­nalizable, show that A-1 is orthogonally…

A:

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

A:

Q: Consider the given matrix. Determine det(A) and the inverse matrix of A.

A: Here we have to find the determinant and inverse of a matrix.

Q: Let A be an m×n matrix with rank m and B be an n×p matrix with rank n. Determine the rank of AB.…

A: This can be done as follows:

Q: If A is an invertible matrix, explain why A and AT must have the same null space. Is this true if A…

A: If A is invertible matrix then it has trivial null space and a matrix is invertible iff AT is…

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

A: It is given that AB is invertible, that means there exist some matrix C such that:(AB)C = C(AB) = I…

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: Prove that any square matrix A can be written as the sum of a symmetric matrix and a skew­symmetric…

A: To Prove Any Square matrix A can be written as the sum of a symmetric matrix and a skew­symmetric…

Q: = Use spectral decomposition to orthogonally diagonalize the matrix A [] -7 24 7 24

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Q: "Let A, B be n x n matrices. If 3A(B“) is invertible, then both A and B are invertible."

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Q: VýB × VA

A:

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: Show that if A is a symmetric matrix, then Ak , k = 2, 3,... , is symmetric.

A: Given that, A is a symmetric matrix. Then we get, A^T = A; where A^T is the transpose of matrix A We…

Q: Find a matrix P such that P'AP orthogonally diagonalizes A. Verify that P'AP gives the proper…

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Q: Let X be any n x k matrix. Show that X'X is a symmetric matrix.

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Q: Let X Give an example of two 2 × 2 matrices A and B, neither of which is the identy matrix I, such…

A: Ok

Q: If A and B are invertible matrices, show that AB and BA are similar.

A:

Q: Find a spectral decomposition of the matrix in the given exercise.

A: The given matrix is

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

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

Q: 8. Let A be a matrix 4 0 0 0 0 4 0 0 0 0 2 0 1 0 0 2 Determine if A is diagonalizable, and if…

A: Matrix A is diagonalizable if there exists an invertible matrix P and a diagonal matrix D, such that…

Q: Let A and B be two square matrices. Assume that both A and B are invertible. Show that BA is…

A: Note that--- If determinant of a matrix is non-zero then the matrix is said to be invertible and if…

Q: Let A and B be two skew-symmetric matrices of the same size. Show that AB+BA is symmetric.

A: Given that: Let A and B be two skew-symmetric matrices of the same size. To show: AB+BA is…

Q: Let A be a matrix 4 0 0 0 0 4 0 0 0 0 2 0 1 0 0 2 Determine if A is diagonalizable, and if…

A: Solving the above matrix problem using row operations

Q: Suppose A is a matrix such that A² + A = I, I is the identity matrix. Show that a) A must be square.…

A:

Q: Determine whether the matrix is orthogonal. An invertible square matrix A is orthogonal when A−1 =…

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Q: Let A and B be two square Matrices of the same Size such that A is not invertible (B may or not be…

A:

Q: Consider the multiplication AB is existing with the size (M × N) , where B is a matrix of size (L ×…

A: Consider the given information. The multiplication of two matrix is possible if and only if the…

Q: Show that the matrix 2 1 A = 1 3 2 1 1 1 is invertible, but that A cannot be written as the product…

A: Solve the given problem.

Q: Let A be an m x n matrix. Is ATA symmetric? Explain.

A: Given that, A is an m×n matrix

Q: Let A be a 3 x 3 matrix such that 3a(1)- 2a(2)+7a(3)=0. Is A nonsingular? Explain.

A: A is singular matrix if rank is not full

Q: 5. Use spectral decomposition to orthogonally diagonalize the matrix A = 5 10 {]. 5

A:

Q: If A is an invertible matrix that is orthogonally diagonalizable, show that A-1 is orthogonally…

A: Let the orthogonal decomposition of A be denoted as A = QTDQ where Q is orthogonal and D is…

Q: Let A be an m × n matrix. Explain why the matrix multiplications ATA and AAT are possible.

A:

Q: Let A be a 3 x 3 matrix such that A4 - 3A³+ 4A²- 613 = 0. Then, A is invertible and

A: Here we have to use the concept matrix.

Q: Let A be a nonsingular matrix. Show that A−1 is also nonsingular and (A−1)−1 = A.

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Q: The multiplicative identity matrix of order 2 is I2 =_______.

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Q: Let U and V be n × n orthogonal matrices. Explain why UV is an orthogonal matrix. [That is, explain…

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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 -1 1 A = -1 1 -1 -3 3 Find an upper triangular matrix which is orthogonally similar to A.

A: An upper triangular matrix is a triangular matrix with all elements equal to below the main…

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 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: APPLY A SERIES OF ROW OPERATIONS ONTO THE GIVEN MATRIX "A" TO FIND ITS INVERSE. [ A|1] => [ I|…

A:

Q: If AC is an invertible matrix, then show that A is also an invertible matrix

A: Given- AC is an invertible matrix, i.e.- ACAC-1=I⇒ACA-1C-1=I⇒AA-1CC-1=I

Q: Let A and B be similar invertible matrices. Show that the matrices A-1 and B-1 are similar.

A: If A and B be similar invertible matrices then to show that the matrices A-1 and B-1 are similar.

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.