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

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 matrix A is diagonalizable, then AT is diagonalizable.

A: We have given a matrix A, that is diagonalizable.

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 if the entries in each row of an n × n matrix A add up to zero, then the column vectors of…

A: linearly dependent means det of A is zero

Q: Find x such that the matrix A is equal to its own inverse.

A: We have A=-2-33x Now, A-1=1AadjA For a 2×2 matrix A=abcd, we have A=ad-bc  , adjA=d-b-ca Hence, for…

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: The inverse of a matrix A is written as A- such that AA-1 =A-A=

A: Clearly we know that, AA-1 = I.

Q: .

A:

Q: Assuming that b # 0, find a matrix that orthogonally diagonalizes a= [: :] P

A:

Q: Let x1, x2, and x3 be linearly independent vectors in R4 and let A be a nonsingular 4 × 4 matrix.…

A: Given that:

Q: Prove that the rank of a matrix, every element of which is unity is 1

A: Rank of a matrix refers to the number of linearly independent rows or columns of the matrix.To find…

Q: If A can be row reduced to the identity matrix, then A is not invertible. True False

A:

Q: If the rows of a square matrix A are linearly dependent then A is singular Select one: O True O…

A:

Q: Let A be a 3 × 3 matrix and let x1, x2, x3 be vectors in R3. Show that if the vectors y1 = Ax1, y2 =…

A: Let A be a 3 × 3 matrix.

Q: Prove that for any square matrix A, ||A||3 E ||A* A||2.

A: For any square matrix A Required to prove A22 <=A*A Let, us consider the matrix, A=2314

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

A: Matrix A is said to be non singular if the determinant value is non zero.

Q: Show that the column vectors of following matrix A are linearly independent:

A: the given question is taken from the matrix in which we have to find show that the given matrix A is…

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

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

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: Show that if A is a symmetric nonsingular matrix then A−1 is also symmetric.

A: A is a symmetric non singular matrix, which means inverse exists and

Q: If A and B are square matrices of the same order, then (AB)T = AT BT True False

A:

Q: Find three values of c for which the matrix C is not invertible.

A:

Q: Prove that B is row equivalent to A if and only if there exists a nonsingular matrix M such that B =…

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: Show that if A is an orthogonal matrix, then det(A) = ±1.

A: Given If A is an orthogonal matrix. To show det(A) = ±1

Q: Find the two by two matrix A such that A -

A:

Q: If both matrices A,B are orthogonal prove that (ab) is also orthogonal

A:

Q: Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists).

A: Given:0111 over ℤ2

Q: If A and B are n x n matrices, and vector v is in the image of both A and B, then i must be in the…

A: True or false.

Q: Find the inverse of the matrix A using row operation if A is a non-singular matrix, where, [1 3 -2]…

A: A=13-223001-1

Q: Determine whether the given matrices are linearly independent.

A: Given: 1234,4321 Let  λ11234+λ24321=0λ1+4λ22λ1+3λ23λ1+2λ24λ1+λ2=0

Q: Suppose A and B are matrices with AB=0 (where 0 represents the zero matrix of the appropriate size).…

A:

Q: Let A and B be equivalent square matrices. Prove that A is nonsingular if and only if B is…

A: Given, A and B are equivalent square matrices. Let P be any non singular matrix of the same…

Q: Let U be the row-echelon form of square matrix A. prove that if the first r columns of U are…

A:

Q: Show that if A is an orthogonal matrix, then det(A) = +1.

A: To prove, detA=±1

Q: Determine whether the given matrices are linearly independent.

A: Let a, b, c be scalars such that, a·1243+b·21-10+c·1111=0000 Then we get,…

Q: Show that if a matrix A has an inverse A-, then in fact A-1 has an inverse as well

A:

Q: Let A and B be nonzero n X n matrices such that A is symmetric and B is skew-symmetric. Prove that…

A: As you will encounter in most scenarios when trying to prove that a set is linearly independent,…

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

A:

Q: If A is an n×n matrix and {v1,v2,…,vn}    is a linearly independent set of vectors in Rn, then…

A: This is true

Q: Please see image

A: Given matrices are

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: Prove the following statement: If the columns of A are linearly independent, then AT A is…

A: A is invertible if there exists a matrix A-1 such that AA -1=A -1A=I   The vectors v1,.......,vn are…

Q: If A and B are invertible matrices and c is any scalar, then which of the following does not hold?

A: A and B are invertible matrices then  (AB) -1  = B-1 A-1

Q: Show that if B is a symmetric nonsingular matrix, then B2 is positive definite.

A: Concept: A branch of mathematics which deals with symbols and the rules for manipulating those…

Q: Show that if A is an orthogonal matrix, then AT is also orthogonal.

A:

Q: Construct a nonzero 3 × 3 matrix A and a nonzero vector b such that b is in Col A, but b is not the…

A: We will construct A and put b as a linear combination of column vectors in A.

Q: Suppose matrix A is m×n and B is n×p. Prove that if the columns of B are linearly dependent, then…

A: Let A be m by n matrix and B be a n by p matrix.

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

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

Q: If A is any invertible n x n matrix, then rref( A) In. %3D True False

A: Invertible matrices, An n×n matrix A is invertible if there exists an n×n matrix B such that…

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.