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. Conversely, show that if A is any 3×3 matrix having rank 1, then there exist a 3 x1 matrix B and a 1 ×3 matrix C such that A = BC.

Q: For a nonsingular matrix A and a nonnegative integer p, show that (Ap)−1 = (A−1)p.

A: The objective is to show that AP-1=A-1P for a non singular matrix A and a non negative integer P

Q: Let A, B be 2 x 2 matrices, C a 3 x 3 matrix and D a 1 × 1 matrix. Find the condition on these…

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Q: Find conditions on a, b, c, and d such that B = as attached commutes with every 2X2 matrix

A: B=abcd Let A=uvxw BA=au+bxav+bwcu+dxcv+dwAB=ua+vcub+vdxa+wcxb+wd Let BA=AB…

Q: If A is an upper triangular 3x3 matrix whose diagonal entries are all zero, prove that A' = 0.

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Q: If B is a matrix obtained by multiplying each entry of some row of a 2 × 2 matrix A by the scalar k,…

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Q: For any 3x3 matrices A & B: Verify that (AB)" = B" A".

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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: Prove that if A is an n X n matrix, then A - Ar is skew-symmetric

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Q: Given a matrix C, Shows that by using Kronecker Product: C ⊗ (B + A) = C ⊗ B + C ⊗ A

A: The Kronecker Product of the matrix A∈Mp,q with the matrix B∈Mr,s is defined as…

Q: nd an invertible 2 x 2 matrix P and a 2 x 2 diagonal matrix D such that P-¹ AP = D, and determine…

A: To solve the following problem on diagonalization given eigen values and eigen vectors.

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: Given that A, B, C are all square matrices of the same size, matrices A and C are both orthonormal,…

A:

Q: Let A be a symmetric positive definite n×n matrix. Show that A can be factored into a product QQT ,…

A: We have given an n×n matrix A that is symmetric positive definite; We have to show that A can be…

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

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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: Create a 5 x 3 matrix, call it A, with no more than 3 zeros such that the columns of A are linearly…

A: Let us consider a matrix A=-4-300-14103546-7-3-9 First we will show that columns of matrix are…

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

A:

Q: Suppose B = QRQ-1, where Q is orthogonal, and R is upper triangular. Show that if B is symmetric,…

A: Given that B=QRQ-1 where, Q is orthogonal and R is upper triangular. Then, multiply Q on both sides:…

Q: Suppose A is a 2 x 2 matrix such that A and 1 A What is A-1?

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: Let A be an m x n matrix with m>n.  Show that for any n x m matrix B, then AB does not equal Im. Use…

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Q: If A and B are square matrices of the same size such that AB = BA, prove that (AB)2 = A2B2.  By…

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Q: Let A be a m × n matrix with columns c1, …, cn. Prove that {c1, …, cn} is independent if and only if…

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Q: Use the Cayley-Hamilton theorem to find A1, A°, and A“ for the given matrix A. 10 -8 - 4

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Q: Show that if A is any m × n matrix, then AAT and AT A are symmetric.

A: There is a matrix  A given of order m×n. To show that AAT  and  ATA  are symmetric. Solution: The…

Q: If A is an m X n matrix and B is an n x p matrix, matrix. To find in matrix A then AB is defined as…

A:

Q: Suppose a matrix is given as X Y A where X is a non-singular m x m matrix, Z is a non-singular n x n…

A:

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: Construct a 3 × 2 nonzero matrix A such that: Ax = 0 has only the trivial solution.

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

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

A: Given that A is nonsingular matrix. That is, A≠0. Since k is a non zero scalar, kA≠0. Therefore, kA…

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: Prove that, for square matrices A and B, AB = BA if and only if (A - B)(A + B) = A2 - B2

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Q: Show that if A is any m × n matrix, then Im A = A and A In = A.

A: Given: Matrix A of order m × n.

Q: Prove that if a row or a column of an n × n matrix A consists entirely of zeros, then det(A) = 0.

A:

Q: Suppose A and B are NONZERO matrices such that AB = [0]. Then there is no other matrix C such that…

A: We have to check whether given statement is true or false.

Q: Let A and B be n × n matrices and assume that columns of B are linearly dependent. Show that columns…

A:

Q: (a) Prove that if A and B are symmetric n X n matrices, then so is A + B. (b) Prove that if A is a…

A: Part-(a) : Given : A and B are symmetric n×n matrices. We have to prove that A + B is symmetric n×n…

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: Show that if an n × n matrix A is positive definite, then there exists a positive definite matrix B…

A: Consider the provided question,

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

A: To show : A is nonsingular if and only if B is nonsingular.    Given A  and B are two row equivalent…

Q: Show that if a square matrix A has two equal columns, then A is not invertible.

A: since not a particular question asked ,as per guidelines solution to only first question (i.e Q1) is…

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: Let A and B be n × n matrices. Show that if AB is non- singular, then A and B must be nonsingular.

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Q: Use the theorem given below to show that if two rows of a square matrix are equal, then det A = 0.…

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: 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: 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.