Find the rank and nullity of the matrix; then verify that the values obtained satisfy Formula 4 of the Dimension Theorem

Q: Find two 3x3 invertible matrices A and B, such that A+B is not invertible.

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Q: Suppose that A is a 18 x 7 matrix and the null space of A has dimension 3. What is the rank of A?

A: For a m×n matrix, the number of rows is equal to m and the number of column is equal to n .

Q: Prove that if A is invertible and B is row equivalent to A, then B is also invertible.

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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: - 1 1 1 1 7 - 7 - 3 -3 6 4 6 -2

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Q: 32. Construct a nondiagonal 2 x 2 matrix that is diagonalizable but not invertible.

A: .

Q: The multiplicative identity matrix of order 3 is I3 =_______.

A: Let the Identity matrix be…

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

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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: Given a particular matrix A and a vector x + 0, discuss how to determine whether x is an eigenvector…

A: Let A is any matrix of order n×n. The scalar λ is an eigen value of A when there is non zero vector…

Q: Find the matrix C such that BC = AT

A: Let A be an n×n matrix, Then the transpose matrix of A is also an n×n matrix denoted by AT. Now,…

Q: Find an orthogonal basis for the column space of the matrix to the right. - 1 7 3 -6 4 2 -1 1 -5 -2

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Q: Find (a) a basis for the row space and (b) the rank of the matrix. [0  1 −2]

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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: Find the standard matrix of the linear operator M: R2 R2 that first reflects every vector about the…

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Q: State Cayley-Hamilton theorem and prove this for following matrix [1 1 A 1 2 -1 %3D [5 8

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Q: Construct a 3 x3 nonzero matrix A such that the vector is a solution of Ax = 0. 1 LA =

A: x=-1-1  1 Now let A=abcdefghi be the required vector. Then, Ax=0⇒abcdefghi-1-1  1=000⇒-a-b+c-d-e+f…

Q: Let A and B be n × n matrices such that AB is invertible. Show that A must be invertible. You cannot…

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Q: Prove that the main diagonal of a skewsymmetric matrix consists entirely of zeros.

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Q: If the rows of an n X n matrix A are linearly inde­pendent as vectors in Rn, what is the rank of A?…

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Q: Find Jordan form of this matrix

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Q: Prove that if a square complex matrix is unitarily diagonalizable, then it must be normal.

A: Every square complex unitarily diagonalizable matrix is normal. Let, A be a square complex unitarily…

Q: Find the standard matrix for the linear operator T: R^3 --> R^3 that first rotates a vector…

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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: If an nxn matrix is invertible, then the column vectors form a basis for R

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Q: Due to what property, applied to the matrices and scalars below, do we know that c(A + B) = cA + cB…

A: Given: We have to state which property for c(A + B) = cA + cB and (c + d)A = cA + dA

Q: Find (a) a basis for the row space and (b) the rank of the matrix.

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Q: If A is an m × n matrix of rank r, what are the dimensions of N(A) and N(AT )? Explain.

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Q: If a matrix A E Mn(R) and rank(A) = n, is it true that the column vectors of A are linearly…

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Q: Show that the diagonal entries of a Hermitian matrix must be real.

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Q: FIND THE FUNDAMENTAL MATRIX AS W ELL AS THE FUNDAMENTAL MATRIX SATISFYING )- I x'= -4 %3D

A: For the solution follow the next steps.

Q: Let the n × n matrix A be equal to AT . Briefly describe the pattern of the entries in A.

A: Consider the matrix A of order n×n such that AT=A. let, A=a11a12…a1na21a22…a2n⋮⋮⋱⋮an1an2…ann  here…

Q: If the null space of a 7 × 6 matrix  is 5-dimensional, what is the dimension of the column space of…

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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 the null space of a 7 x 6 matrix A is 5-dimensional, what is the dimension of the column space of…

A: We solve given problem with the help of rank nullity theorem

Q: The size of the matrix X'Y, where X is a design matrix of size (k+1) by (k+1) and Y is the vector of…

A: The size of design matrix X is given by (k + 1) by (k + 1)  And Y is the vector of values of…

Q: If A is a square matrix whose rows add up to the zero vector, explain why A cannot be invertible.

A: Given A is square matrix whos rows add up to zero vector.

Q: 1 2 3 A = 1 3 2 2 6 8

A: See solution below

Q: 3) A = [3 Calculate the 10th power of the matrix with the Cayley-Hamilton Theorem.

A: Given:

Q: Is it true that when an n*n matrix whose columns form a basis of R^n, then A is invertible?

A: We shall answer this in next step

Q: Determine the matrix C such that (CT – 21) 1 1 2 2 3

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Q: Let A and B be n x n matrices and c be a scalar. Prove that, if B is obtained by multiplying a row…

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Q: Generalize the result by proving that, for any comfortable non-singular matrices A,B and C the…

A: Consider the given information. ABC-1=C-1B-1A-1 Multiply by pre and post C.…

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: Every matrix projects orthogonally any vector in its domain to its column space. O TRUE O FALSE

A: The given statement where the matrix suppose M , then the projection of any vector say 'z'…

Q: Show that the condition number of the square matrix, X (generated with X;j = 461-2), grows i+6j–2 as…

A: We will use the conventional definition of condition number of a matrix.

Q: Show that it is impossible to find two real 2 x 2 matrices A and B such that AB - BA = dI where d…

A: Take A and B are arbitrary matrices and find AB-BA and analyse the diagonal elements

Q: Find a formula for the dimension of the vector space of symmetric n X n matrices.

A: Given: symmetric  n×nmatrix