Prove that if A is an 2021 x 2021 skew-symmetric matrix, then A must be a non-invertible matrix.

Q: If A be an n x n invertible matrix. Prove that the transpose of A is also invertible.

A: A square matrix is invertible matrix if its determinant is nonzero. To obtain the transpose of the…

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 one row (column) of the n × n matrix A consists entirely of zeros, then A is singular.

A: We know that if all entries of one row(column) of the n×n is zero then determinant of matrix is…

Q: Let A be a m×n matrix. Prove that AB and Ba both exist only if B is an n×m matrix.

A: Explanation of the answer is 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: Suppose A and B are n×n matrices such that B is invertible and AB is invertible. Prove that A is…

A: Given that the matrices A and B are of order n×n and both A, AB are invertible.

Q: prove that if A and B are invertible matrices with the same size then AB is also invertible and :…

A: A and B are invertible. So, AB=BA=I

Q: Show that an n x n nonnegative matrix is transitive if and only if £=1 4* is a positive matrix. -k=1

A: Given that, An nonnegative matrix is transitive.…

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: Show that if A is a symmetric matrix, then AT is symmetric.

A: we have to show that if A is a symmetric matrix, then AT is symmetric.   as we know that a matrix B…

Q: Prove that if A is an invertible n × n matrix, then the rows of A span R“ by using transpose.

A: we use the fact, if A is invertible then A is row equivalent to the identity matrix I.

Q: IF A is a nonzero n X n matrix such that A 0, then A is singular. %3D Select one: True False

A:

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: 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: 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 E is an elementary matrix if and only if Et is.

A:

Q: Show that if A is a nonsingular matrix, then adj A is non- singular and 1 (adj A)- = A = adj(A-).…

A: Given: A is a nonsingular matrix. Since A≠0

Q: Prove If A is an invertible matrix and c is a nonzero scalar, then cA is an invertible matrix and…

A:

Q: If A is a symmetric matrix, then AT =

A: If  A is symmetric matrix then AT is also symmetric matrix

Q: Let A be a symmetric matrix and P be an orthogonal matrix. Show that A is positive semi-definite if…

A:

Q: Show that for an invertible matrix A: 1 det (A-1) det (A)

A: Given that, the matrix A is invertible. It is known that, detI=1 and detA·A-1=detI.

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: Prove If U is a unit upper triangular matrix, then U is nonsingular and U−1 is also unit upper…

A:

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: 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 and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.

A: If A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.

Q: prove that if A is an invertible matrix then AT is also invertible and (AT)-1= (A-1)T

A: Assume a matrix B. Such that B=A-1

Q: Prove that if the diagonal entries of a diagonal matrix are permuted, then the resulting diagonal…

A: Permuting the diagonal entries of a diagonal matrix D amounts to permuting the rows of that matrix…

Q: Prove that with gauss jordan! If A is an n squared matrix and if we can find a matrix B such that…

A:

Q: Prove that for a diagonal matrix, the 1-,2-,∞- and -norms are the same.

A: We are given a diagonal matrix D. We wish to show that D1=D2=D∞ If we write D=d10⋯00d2⋱⋮⋮⋱⋱00⋯dn…

Q: If A be an n x n invertible matrix. Prove that the inverse of A is unique.

A: The assumption for the matrices helps to prove the required. A matrix is said to be invertible if…

Q: If A is a square matrix such that A = A1, then A must be equal to either the identity matrix I or…

A:

Q: Prove that an upper or lower triangular n x n matrix is invertible if and only if all its diagonal…

A: Consider A be a n x n upper or lower triangular matrix

Q: Let A be a square matrix such that A2 = 0. Prove that (Af)2 = 0

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

Q: Prove that if A is a symmetric matrix, then adj A is symmetric

A: Let A be A symmetric matrix.

Q: Prove that if A is an invertible matrix then det(A−1) = 1/det(A)

A:

Q: Let A be an n × n symmetric, positive definite matrix. Show that if P is any n × n nonsingular…

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

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Q: An n x n matrix A is invertible if ВА In АВ II

A: Inverse Matrix: Let A be an n × n matrix. If there exists an n × n matrix B such that AB = I = BA,…

Q: Prove that if A and B are square matrices and AB is invertible, then both A and B are invertible.

A:

Q: Prove that if a symmetric matrix is invertible, then its inverse is symmetric also

A: Let A be a invertible symmetric matrix. To prove that, Inverse of a symmetric matrix is symmetric.…

Q: 3. Determine whether A is diagonalizable and, if so, find an invertible matrix P and a diagonal…

A: Hi, since you have asked multiple questions, we will solve the first question for you. If want any…

Q: Let B be an n x m matrix with rank m. Prove that there exists an m x n matrix A such that AB = Im.

A: Consider the given matrix n×m with rank m

Q: Let A be a symmetric matrix and P be an orthogonal matrix. Show that A is positive semi-definite if…

A:

Q: Let A be an m x n matrix. Prove that if n is an odd number, then rank(A) = nullity(A).

A:

Q: Prove that the symmetric matrix is diagonalizable.

A: Concept: A rectangular array of numbers (or other mathematical objects) for those the operations…

Q: Prove that if A is an orthogonal matrix, then ∣A∣ = ±1.

A: Given: A is orthogonal matrix   To Show: A=±1