10. Suppose that A is an n x n matrix with real entries such that the diagonal elements are all positive, the off-diagonal elements are all negative and the row sums are all positive. Prove that det A# 0.

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

A: For any n×n matrix A, ATA=AAT

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…

A:

Q: 6. If A is an nxn matrix whose LU decomposition is PA = LU, , prove that det(A) = (-1)' u₁₁ U 22.²…

A:

Q: Suppose that A is an n x n matrix such that A² = 0. a) Show that if 7 E im(A), then 7 € null(A). (In…

A:

Q: Prove that if A is an m x n matrix of rank r, then the dimension of the solution space of Ax = 0 is…

A:

Q: 2. Suppose A is a square matrix. Which matrix is not symmetric? O A+AT O AAT ОА-АТ O ATA

A: Note the following results if A and B are two matrices ABT=BTAT…

Q: How is the matrix of R^−1 obtained from that of R in general, that is, when A = {a1, . . . , am}, B…

A:

Q: 6. Suppose  X + XB = C, where X, B, and C are square matrices of the same size. a.  Derive an…

A: We will see that (I+B) must be invertible for existence of X.

Q: 9. Suppose that M is a 2 × 2 matrix over R, with characteristic polynomial det(AI – M) = X² – 21 + 3…

A:

Q: 2) An n x n matrix A is called nilpotent if Ak = 0 for some positive integer k. Show that if A + 0…

A:

Q: highlighted" condition. A matrix in M2(R) which is “nilpotent" and has a non-zero entry in each row…

A: A matrix is Nilpotent if Ak = 0 where 0 is null matrix and k is index of matrix A.

Q: 2) Suppose that A and B are nxn matrices that satisfy AB=AB. Show that AB = AB.

A:

Q: TRUE/FALSE Assume that all the given matrices are n × n. If A = O, but Ak = O (where O denotes the…

A: Given:   To Check: A is singular or not Concept: A is singular only when det(A) = 0

Q: Suppose A and B are nxn matrix and P is an invertible nxn matrix such that A = P-BP (a) Show that A™…

A: We will use Diagonalization  and proof of first part to find power of matrix

Q: 2. Suppose that A is a 64 × 17 matrix and that the rank of A is 11. What is the dimension of the…

A: The given matrix is A of size 64×17. Also the rank of A is 11. As we know that for any matrix of…

Q: Let A be an m × n matrix. Under what circumstances will ||A||2 = ||A||F?

A:

Q: Let A be a nonsingular n×n matrix with a nonzero cofactor Ann, and set c = det(A)/Ann Show that if…

A: Let A is any non singular nxn matrix with non zeroCofactor Ann and Set C = det(A)Ann   By concept…

Q: Find a matrix A such that [is: is a solution for AX=X'.

A:

Q: [:] Suppose A is a 2×3 matrix such that A and A -3 Compute the product A

A: I have writen the vectors in linear combination of given two vectors and applied A in both sides

Q: Suppose that A+ is a 2 × 2 matrix and that A+ = -A. Show 7.2 that iA is awesome.

A:

Q: 6. If A is an nxn matrix whose LU decomposition is PA=LU, prove that det(A)=(-1)' u₁22... Un where r…

A: Given: A is an n×nmatrix, whose LU decomposition is PA=LU.   To show: detA=-1ru11u22...unn, where r…

Q: Suppose that A is a 5 x 5 matrix which has a null space of dimension 4. The rank of A is rank(A) =

A:

Q: 4. Suppose A is a nonsingular matrix with SVD A= UEV*.

A:

Q: 5. Show that if M is a 3 x 3 magic square with weight w, then we can write M as M = Mo + kJ %3D…

A: Given that ,  A       magic square  To show that : We can write as                     Where,  is a…

Q: Let A be an n × n matrix in which the entries of each row sum to zero. Find det(A).

A:

Q: A matrix B in Mnxn (C) is called skew-symmetric if Bt= -B. Prove that if B ϵ Mnxn (C) is…

A: Let B be skew-symmetric matrix of order n, where n is odd, then we have

Q: 2. Verify Calay-Hamilton theorem for the following matrix: (2 1 -1 A = (3 1 1 0 2

A: Solution:Given        A=21-1310102To findA-λI=0

Q: 7. Suppose U is an m x n matrix and V is an n x m matrix for which In-VU is invertible. Show that…

A:

Q: (1) Suppose the last column of AB consists of all O's but B itself has no column of zeros. Are the…

A: as you are asking multiple questions I have solved first one , please repost with the remaining…

Q: : Show that there are no nxn matrices A and B such that AB – BA= In. & You may cite, without…

A:

Q: Suppose A is a 5x9 matrix and Rank(A) = 3. Find dim(null(A)). 1 O 2 Оз O 6

A: Solution of the given problem is as follows:

Q: Suppose that A is an n × n matrix and det A + 0. Determine the rank and nullity of A.

A:

Q: 4. Prove: If A and B are both n x n matrices, then det(AB)= detAdetB. (Only use 2x2 matrices)

A: Given A and B  n×n  matrices to prove  det(AB)=det A det B

Q: 7. If a n x n matrix B is obtained from A, using only m number of row interchanges, then det A det B…

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: If A and B are singular n × n matrices, then A + Bis also singular.

A: If A and B are singular  matrices the A+B is also singular. (False Statements)

Q: Find a 3 x 3 matrix A such that Ax = 8x for all x in R³. A =

A:

Q: if the column sums of a matrix C are all strictly less than 1, then the inverse of the matrix I-C…

A: Assume that the column sum of the matrix C is strictly less than 1. Then for sufficiently large m,…

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: Verify that A = I when A =. l-8 -1 Use partitioned matrices to show that M² = I when 1 0 |-8 -1 0 0…

A:

Q: 5. Show that if M is a 3 x 3 magic square with weight w, then we can write M as M = Mo+ kJ where Mo…

A: M is a 3 x 3 magic square matrix with weight w. We take weight, w=15 The matrix of M is  Add any…

Q: Suppose that a matrix M is nilpotent if exists an x element of Z+ in where Mx = 0. If M is…

A: We will prove the given result.

Q: 2.3 Compute the principal square root of each of the lowing positive semidefinite matrices. (b) 2 3…

A: We shall solve first question only as you have asked more than one Que of same type. For others…

Q: Show that A and B are not similar matrices. 2 1 , В 3 3 -1 A = -3 7 -6 Since A has characteristic…

A:

Q: 5. Show that the condition that the block matrices (8 ) m ( ) A 0 А С and B В can be matrices of the…

A: Given: The block matrices, A00B and AC0B.   To show: The given block matrices can be matrices of the…

Q: For matrices M and N, the 3rd column of NM is all zeros. Matrix M has 5 columns and none of them are…

A: Please find the detailed solution in next steps

Q: Write down what it means for a n x k matrix to be singular.

A: As we know, If matrix is singular then it has determinant equals to '0' For Finding determinant…

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.