Let A be an n x n matrix. Define A1=1/2(A+A*) and A2= 1/2i(A-A*). a) Prove that A1*= A1, A2*=A2, and A=A1+iA2. Would it be reasonable to define A1 and A2 to be the real and imaginary parts, respectively, of the matrix A ? b) Let A be an n x n matrix. Prove that if A=B1+iB2, where B1*=B1 and B2*=B2, then B1=A1 and B2= A2.

Q: Prove or disprove: For any n×n matrix A, AT A = AAT

A: For any n×n matrix A, ATA=AAT

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 = [aj be a 4 x 4 matrix. Develop the general expression for det(A).

A: Given:               Matrix A is : [aij] is the 4×4 matrix;   General expression for the determinant…

Q: Let A be an m × n matrix. Prove that N(A) ⊂ N(ATA).

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

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Q: let Amxn matrix , c scalar If cA=0, then Select one: a. c=0 and A=0 b. c=0 or A=0 C. C0 or A=0

A: WE have to check the meaning of cA=0.

Q: Does there exist an 8 x 8 matrix A - (a) satisfying the following three conditions? (1) If i 2j then…

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Q: Let A be square matrix. If A" = 0. A= Oava %3D for n> 2. Then

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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: 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 a real n xn matrix, show that AA" and ATA are similar. (Hint. Start with an SVD for A.)

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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: a) Show that there is no matrix A E M250x250 with real entries satisfying AAT = -I.

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Q: Is it true or false? If A is nxn matrix, c is scalar, then cA |=c | A| Birini seçin:

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Q: Consider an n x n matrix. Give 3 equivalent conditions for A to be invertible

A: Let A be an n×n matrix. we have to give 3 equivalent condition for A to be invertible. Conditions…

Q: .

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Q: Find three non-trivial 2 x 2 matrices such that the square of the matrix is the zero matrix, i.e: A?

A: We want to find matrix A such that A2 = 0000

Q: Prove that a matrix A is positive definite if and only if it can be written as A = RTR, where R is a…

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Q: Show that if A is a mxn matrix and rank(A)=m then Ax=b has a solution for any vector b in Fm (Hint:…

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Q: Find x such that the matrix A is nonsingular.

A: Given: The inverse of a matrix is given as follows,

Q: Show that if A exists for an n x n matrix A, then A- is unique.

A: Since, you have posted multiple question but according to guidelines we will solve the first…

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 be and m x n matrix. Describe the matrix E such that EA is A with row swapped.

A: Given matrix A is a m×n matrix. And the elementary matrix E and EA are row swapped. Since elementary…

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 n × n matrix and let x and y be vectors in Rn. Show that if Ax = Ay and x /= y, then the…

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Q: Let A be an nxn matrix and B = A+1, where I is the identity matrix, then it is possible for A and B…

A: Given A be an n×n matrix nd B=A+I where I is the identity matrix , then it is possible for A and B…

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

A: Given that, A is an m×n matrix

Q: Show that if A is a symmetric positive definite matrix, then A is nonsingular and A−1 is also…

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Q: Let A, B, C be invertible matrices of order 3x3. Show that if AB = AC , then B =C. %3D

A: Let A, B, C be invertible matrices of order 3×3. To show that if AB=AC, then B=C.

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

A: Given, A is an upper triangular matrix

Q: Construct a 3 x 3 nonzero matrix A such that the vector is a solution of Ax = 0.

A: Given conditionAx = 0          .................(1)x = 111 is a solution to Ax=0,Let matrix A be - A…

Q: Find x such that the matrix is singular.

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Q: For a nonsingular matrix A and nonzero scalar k, show that (kA)-1 = A-1.

A: For a nonsingular matrix A and nonzero scalar K To show that kA-1=1kA-1 Consider a non singular…

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: 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: Determine the matrix C such that (CT – 21) 1 1 2 2 3

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Q: Show that if A is an n × n matrix, then det(AAT ) ≥ 0.

A: Given, A is a n x n matrix. Let AT be the transpose of A. To prove det (AAT) is non-negative.

Q: Let A and B be n x n matrices and c be a scalar. Prove that, if B is obtained by multiplying a row…

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Q: Prove that if A is an invertible matrix and B is row equivalentto A, then B is also invertible.

A: Recall the following equivalent statements as follows.

Q: Let A be an n × n matrixSuppose that the matrix C is obtained from A by interchanging the ith and…

A: Matrix C is obtained from A by interchanging the ith and jth row of A. We need to find an elementary…

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: Let A be an m × n matrix.(a) Describe the diagonal entries of AT A in terms of the columns of A.(b)…

A: (a) Describe the diagonal entries of AT A in terms of the columns of A.

Q: Let Q be an orthogonal matrix such that QA makes sense.                          Show that (QA)+ =…

A: Given, A be a matrix and Q be an orthogonal matrix such that QA makes sense. We have to use product…

Q: Let A = [a;;] be the n x n matrix defined by aji = k and a;; = 0 if i + j. Show that if B is any n x…

A: Given: A=aijn×n,     aii=k and aij=0 if i≠k. Let B be a square matrix of order 3 which can be…

Q: The mxn zero matrix , denoted by 0mn or simply 0 , is the matrix whose elements are all zero , find…

A: A matrix is a mathematical object containing numbers arranged in rows and columns. The matrix in…

Matrices

‘Matrices’ is the plural form of the word matrix, and it is basically a spreadsheet in the form of a box. In mathematics, various functions can be carried out with matrices. Generally, a matrix comes in the shape of a square or rectangle. The elements are distributed horizontally in the form of rows, and vertically in the form of columns.