Let the entries of a matrix A = \aij be defined as aij 2i2 – i+j+ g(j), where g is a %3D function of j. If A is a symmetric matrix, what is g(j)?

Q: Suppose A = PRP–1, where P is orthogonal and R is upper triangular. Show that if A is symmetric,…

A: According to the given information suppose, It is required to show that if A is symmetric then R is…

Q: Let Q be an orthogonal matrix such that QA makes sense. Show that A and QA have the same singular…

A: Let,

Q: Show that there is no matrix A E M250x250 with real entries satisfying AAT = -I.

A: Given that A is real ( All entries are real)  square matrix of order 250×250. The trace of any real…

Q: Show that no 2 x 2 matrices A and B exist that satisfy the matrix equation [1 0 0 1 АВ - ВА. %3D

A: In this question we have to prove that no 2×2 no matrices the given matrix equation.

Q: Let A be a lower triangular matrix with elements aij. Then aij = 0 when O izj O isj O i> j O i<j

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Q: Find an LUfactorization of the given matrix.

A: To Determine: find an LU factorization of the given matrix  Given: we have a matrix12-3-1…

Q: 2x1- 3x2+ 2x3= 9 3x1- 2x2+ X3 = 4 X1 - 4x2+ 2x3 = 6

A: Given system of linear equations is ,                        2x1-3x2+2x3= 9…

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

A: Let the Identity matrix be…

Q: Give an example of a 2 x 2 matrix A such that im(A) =ker(A). %3D

A: Null space of a linear transformation: T:V(F)→V'(F) be a linear transformation then the null space…

Q: Let C be a 5 x 7 matrix with a rank 4. (1) What is the dimension of the solution space of Ax 0?

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Q: Find an LU-factorization of the matrix A such that A = LU. %3D 6 4 -2 1 -2 3 2. -2

A: Explanation of the answer is as follows

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Q: [2ß ß Find all the values of the parameter ß for which the matrix is 3 5B] Consider invertible.

A: The matrix is given as , A=2ββ35β find all values of β for which the matrix is invertible.

Q: Let A = [aij] given by a x bij = (i-j)^3 is a: * Anti-Symmetric matrix Symmetric matrix Identity…

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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=[aij ] be an m xn matrix and k be a scalar then kA is equal to : * [kaij]mxn [aij/k]mxn…

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Q: What is 'a', if B as shown below is a singular matrix?

A: A singular matrix is a matrix whose determinant is zero.

Q: Find a 3 x 1 matrix x with entries not all zero such that 1 2 -1 Ax = 1x, where A 1 1 -4

A: A=1  2-11  0  14-4  5 Now let x=abc be the required matrix. Now consider, Ax=1x⇒1  2-11  0  14-4…

Q: Show that if annxn matrix U satisfies (Ux)· (Uy) = x•y for all x and y in Rn, then U is an…

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Q: 2) If IAI = 3, where A is a 3x3 matrix, caleulate the folowing: 2a) det ( ZA) 26) det (A) 2c) det (…

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Q: Find a (2x2) singular matrix A that maps (3, 1) into (2, 4).

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Q: Let B be an invertible matrix. Show that A = BTB is positive definite.

A:

Q: suppose A be any Hew of have symmetric matoir order and this matrix 4x4 rank of 3 the caleulata the…

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Q: Find the value of x so that the matrix is not invertible. 1. 3 2 0 [ 1 2 1 If answer is in decimal,…

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Q: Construct the matrix with rank one that has Av = 12u for v = ½(1, 1, 1, 1) and u= ½(2, 2, 1). Its…

A: Given: Let A be a given matrix. rank (A) = 1 To find: The only one singular value for  Av = 12 u .

Q: Let A be 5x1 matrix. Knowing that Rank (A) = 2. Then Nullity(A) is equal to: O 2 ONone of these

A: The solution is given below in the next step:

Q: A real square matrix Q is called orthogonal if QTQ = I. Prove det(Q) = ±1. Give an example of such a…

A: Solution

Q: Let A = [0 1 0 0 ], A^(-1) is equal to: * Does not exist Identity matrix Null matrix

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Q: (b) By replacing A with another matrix B different from A in only a single entry, define an inner…

A: To prove a function to be an inner product, we have to show it satisfies the following five…

Q: Let A, B, C be invertible matrices of order 3x3. Show that if AB = AC , then B =C. %3D

A: Let A, B, C be invertible matrices of order 3×3. To show that if AB=AC, then B=C.

Q: Let A be 3 x3 R given matrix by over 2. A = -1 Using the result of A find the value (s) of makes the…

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Q: 18. Let A be a 3 x 5 matrix with 2 pivots. What is the dimension of Nul(A)?

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Q: Let A and B be n x n matrices.  Suppose that AB = On where On is a zero matrix of size n x n, and A…

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Q: 2. Let the rref of a matrix A be, 0 2 0 -2 0 1 -5 0 -3 0 0 0 1 6 Determine A if the first, second…

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Q: TRUE/FALSE Assume that all the given matrices are n × n. If A and B are row equivalent matrices,…

A: Given: All the given matrices are n × n. A and B are row equivalent matrices.

Q: If the ows of a square matrix A are linearly dependent then A is singular

A: To determine whether the given statement is true or false.

Q: Let A be a 6 × 5 matrix. What must a and b be in order to define T : R“ → Rº by T(x) = Ax? Ах?

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Q: Suppose A = UDVT where U and V are n x n matrices with the property that UTU = I and %3D vTV = I,…

A:

Q: Find a 2 × 1 matrix x with entries not all zero such that

A: Given: Ax=3x, where A=2112 let's assume matrix x is x=ab

Q: Find h and k, consistent: which makes matrix Gud Suo = k

A: We have to find the values of h and k for which the given system is consistent.…

Q: Let x and y be square matrices of order n a. Show that if x and y are skew symmetric, then x + y is…

A:

Q: 10 choose a,b,c,d

A: 1. Option a is wrong as the columns of D are LI hence solution will be unique if exists. 2. Option b…

Q: Let A be a nondefective n × n matrix with diagonalizing matrix X. Show that the matrix Y = (X−1)T…

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Q: Let A be a nilpotent matrix of order n then A*n = 0 ) nxA = 0 A =nx1, l is Identity matrix

A: A be a nilpotent matrix of order n

Q: Let A be a lower triangular n x n matrix with nonzero entries on the diagonal. Show that A is…

A: Consider A be a lower triangular matrix with non-zero entries on the diagonal.

Q: Let A be an mxn matrix. Then, the homogeneous equation AX = 0 is consistent if and only if the…

A: Given :- Let A be an m×n matrix. Then, the homogeneous equation AX =O is consistent if and only if…

Q: If A is a skew-symmetric matrix then show that AAT = AT A and A² is symmetric. [Note AT denotes the…

A: We will use definition

Q: The aim of elimination steps in Gauss elimination method is to reduce the coefficient matrix to a)…

A: A matrix where the number of rows is equal to the number of columns is said to be a square matrix…

Q: A matrix of order m x 1 is called a Choose.

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