Prove that if A is an n × n matrix, then A − AT is skew-symmetric.

Q: Given that G is a 1 × 6 matrix and H is a 6 × 1 matrix, a. Is GH defined? If so, what is the order…

A: Given that G is a 1 × 6 matrix and H is a 6 × 1 matrix,a. Is GH defined? If so, what is the order of…

Q: If A is an 7 x 12 matrix, then what is rank A² + dim nul (A²) equal to? Your

A: We can solve this as follows:

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 B is the matrix obtained by interchanging the rows of a 2 ×2 matrix A, then det(B) =…

A: Let the matrix A and the new matrix B be as follows.

Q: Prove that if B is a 3×1 matrix and C is a 1×3 matrix, then the 3×3 matrix BC has rank at most 1.…

A: Let A be the 3 × 3 matrix of rank 1, then it will be in the form as follows.

Q: Let C and D be n x n skew-symmetric matrices. Prove that if CD = – -DC, then CD is a skew-symmetric…

A: Given problem:- Let C and D be n x n skew-symmetric matrices. Prove that if CD=-DC, then CD is a…

Q: Prove that if A is an n × n invertible matrix, then adj(A−1) = [adj(A)]−1.

A: By definition of A-1, we have

Q: 1 0 2 -1 0 2 0 1

A: Given the matrix A=1020-10201 To find an orthogonal matrix P and a diagonal matrix D such that…

Q: If A is a square matrix such that A2 = 0, then A is invertible. True or false

A: A is a square matrix, so, det(AB) = det(A) det(B)

Q: Show that the matrix A given below is singular. A = -2

A: A matrix is singular if it's determinant is zero.

Q: Let A and B be n × n matrices. Prove that if A is nonsingular, then AB is similar to BA.

A: It is given that matrices  A and B are of order (n x n). To show that AB is similar to BA. Since,…

Q: 36) Prove that if A is an invertible matrix and B is row equiva- lent to A, then B is also…

A: You will find solution in next step

Q: Let = Find a nonsingular matrix P and a diagonal matrix D such that P-'AP = D.

A:

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: Let A be an m ×n matrix and B be an n ×p matrix. Prove that AB can be written as a sum of n matrices…

A: Given that A be a mxn matrix and B be a nxp matrix.Now, prove that AB can be written as a sum of n…

Q: Prove that if A and Bare sk ew-symmetric n X n matrices, then so is A + B

A: Given: To Show:   Concept:

Q: Let A and B be m x n matrices and || || be any matrix norm for m x n matrices, prove that | || –…

A:

Q: If A is a nonzero matrix of the form P, then Rank(A) = 2. а

A: Let A be a n×n matrix . If Determinant of A is not ZERO, Then   Rank(A)=n

Q: Prove that if A is an n × n matrix, then det(adj A) = [det(A)]

A: If A is a n×n matrix then A-1=adjA|A|.

Q: Suppose that A is an m×n matrix in which no two columns are identical. Prove that A∗A is a diagonal…

A: Given that A is an m*n matrix in which no two column are identical.Since A*A is defined only if A…

Q: Suppose A is an n×n matrix ,and let v1,...,vn ∈Rn. Suppose{Av1,...,Avn} is linearly independent.…

A: It is given that {Av1,...,Avn} is linearly independent. We will prove that A is nonsingular.

Q: If A is an n X n matrix, prove that det(adj A) = (det A)n-1

A: Given the matrix A of order n×n, and we have to prove that det(adjA) = (detA)n-1

Q: Prove that if A is a symmetric matrix, then PTAP is also symmetric.

A: Given:           A is a symmetric matrix Prove:           PTAP is also a symmetric matrix In linear…

Q: .

A:

Q: (a) Let A be an m × n matrix, and let B be the result of performing a single elementary row…

A:

Q: Prove that if a matrix B has a left inverse then the columns of B are linearly independent.

A: B has a left inverse, that is, there exists an n × m matrix P such that PB = I.

Q: If A is a skew-symmetric n X n matrix and n is odd, prove that det A = 0.

A: Since A is a skew-symmetric matrix we have,  AT=-A where AT is the transpose of matrix A. Then we…

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

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Q: A square matrix is skew-symmetric when AT = −A. Prove that if A is an n × n skew-symmetric matrix,…

A: It is given that A is a skew symmetric matrix.

Q: If a is skew symmetric nxn matrix and n is odd,prove that det A = 0.

A: Consider a skew symmetric n x n matrix A where n is odd. Here the objective is to prove that det(A)…

Q: Show that if A is an n x n symmetric matrix, then |(Ax)•y = x•(Ay) for all x, y in R".

A: In symmetric matrix, transpose of the matrix is equal to the matrix. AT=A

Q: Find a 3 x 3 matrix A such that A = 97 for all in R³. A =

A:

Q: Let A be an invertible, n x n matrix such that A² = A. a) Calculate det(A). b) If n = 3, what is A?…

A:

Q: Show that for any n x n matrix A there exists another n x n matrix M such that MAM-1 = AT, where AT…

A: Consider the given information. Let the matrix A is an n×n matrix and M is a m×m matrix. And the…

Q: Suppose A is a 2 x 2 matrix such that for all x, y E R², (Ax) · (Ay) = x ·y Must it be true that A…

A: Please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: Let A be a nonsingular 4 x 4 matrix such that |A-1| = -2, then AT.adj(A-}) Select one: O True O…

A:

Q: If an nxn matrix A is invertible, then the rowa of A' are linearly independent. T O True O False

A: If a matrix is invertible then its rows are linearly independent

Q: Prove that if A and B are similar matrices, then there exists a matrix P such that Bk = P−1AkP.

A:

Q: Let A be a square matrix such that A2 = O. Prove that (A†)2 = O.

A: Given that A is a square matrix and A2=0.In order to prove (At)2=0 following theorem are…

Q: Prove that if A and B are matrices such that A is symmetric, then (BA−1)T(A-1BT)-1=In

A:

Q: Find a 3 x 3 matrix A such that Aa = 9x for all a in R³

A: We have to find matrix A such that Ax = 9x

Q: O Let A be an m x n matrix, and let B be the result of performing a single elementary row operation…

A:

Q: If A is a 2x2 matrix such that trace(A) = det(A) = 3, then what is the value of trace of (A-!) ?

A:

Q: Prove that if A is an m x n matrix, B is an n x p matrix and C is a p x q matrix, then (AB)C= A(BC).

A:

Q: Show that if A is any n × n matrix, then (a) A + AT is symmetric. (b) A − AT is skew symmetric.

A: Symmetric matrix: Any matrix A is symmetric matrix if and only if it is equal to its transpose that…

Q: (a) Let A be an m × n matrix, and let B be the result of performing a single elementary row…

A: We have to prove that following Statement.

Q: Suppose that A is a 4 x 4 matrix such that det A = 10. Enter the value of det(A³).

A:

Q: Prove that if B is a 3 x 1 matrix and C is a 1 x 3 matrix, then the 3 x 3 matrix BC has rank at most…

A: Let “B” and “C” be arbitrary 3*1 and 1*3 matrices respectively, B=b1b2b3,   C=c1c2c3 Then the…

Q: Prove that if ∣A∣ = ∣B∣ ≠ 0, and A and B are of the same size, then there exists a matrix C such…

A: we have |A|=|B|≠0   A and B are of the same size    We have to prove |C|= and A=CB