If A is a n x n matrix with NON zero determinant, then its columns are a basis for a n dimensional vector space. O True O False

Q: Construct a 3 x 3 triangular matrix A so that 1] the vector 2 is in Col A. 3]

A: We know every lower or upper traingular matrix is a triangular matrix.

Q: t A and B be arbitary n x n matrices. If B is invertible, prove that AB- = B-'A if and only if AB =…

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Q: Let M be a 10 × 10 real matrix such that M2 = M and the determinant of the matrix cannot be 1. Does…

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Q: Let A be an n x n matrix such that A3 = In. (a) Is A invertible? If so, what is A-1? If not, explain…

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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: If the kernel of a 5 x 4 matrix A consists of the zero vector only and if Au = Aw for two vectors v…

A: It is an necessary and sufficient condition for one one function

Q: Consider square matrix B E R":" and identity matrix I E R",". Select all true statements. -1 is If…

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Q: 2. Show that if A is an n x n real symmetric matrix then (Ax) y = x (Ay) for all x, y E R".

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Q: IF A is a nonzero n X n matrix such that A 0, then A is singular. %3D Select one: True False

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Q: 2 If A is 3 x 3 matrix such that A then A is diagonalizable. Select one: True False

A: Assume a matrix A.

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

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Q: The matrix 1 A is an orthogonal matrix. True False

A: Orthogonal matrix- An orthogonal matrix is defined by the matrix which is symmetrical about its…

Q: where En is a k-dimensional vector, A e Ckxk is a constant matrix, and B : Z+ Ckxk is a matrix…

A: Given,      tn+1-t¯=∑i=01αi(tn-i-t¯)+∑i=01βi(zn-i-z¯)…

Q: An n x n matrix A is invertible if and only if it is row equivalent to I. 'n' O True O False

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Q: a If A the matrix A is invertible if and only %3D lc d]° if

A: Find the determinant of the matrix A. A=abcdA=a×d−b×c=ad-bc

Q: .

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Q: If A is an invertible matrix. Then det(AA-): 1D

A: IA is an invertible matrix To find det(AA-1)

Q: Suppose A is (\m \times n \) matrix that has r pivots, then the null space of A contains only the…

A: Option A will be the right answer.

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: If a 4 x 4 matrix A with rows v1, v2, V3, and v4 has determinant det A = 4, %3D 4v1 + 5v2 7v1 + 9v2…

A: See the detailed solution below.

Q: 2. If A is 3 x3 matrix such that A 1 then A is -4 diagonalizable. Select one: True False

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Q: If Aand B are invertible matrices of the same order, Then show that AB is invertible matrices.…

A: LHS   =  (AB)-1  RHS  =  B-1A-1 For    (AB)-1  =  B-1A-1 ,   RHS * inverse of LHS = i…

Q: b) If C is an orthogonal matrix and D = C + C-1 show that D is a symmetric matrix.

A: We have to show D is symmetric.

Q: If P is an invertible n × n matrix, show that: det(P−1A2P) = (det A)2 using the determinant…

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Q: Verify by finding the matrix of S 0 T (a) by direct substitution and (b) by matrix multiplication of…

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Q: If UT = U-1, then matrix U is an orthogonal matrix. %3D O True O False

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Q: IF A is a nonzero n X n matrix such that A = 0, then A is singular. Select one: O True False

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Q: The set B = 4. 2 {[al}- is a basis for Nul(A) for some 4x3 matrix A. Determin

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Q: 0, then A is singular. IF A is a nonzero nX n matrix such that A Select one: True False

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Q: Let A and B be 4x4 matrices such that det(A) = 5 and det(B) = 3. Compute the determinant of…

A: Explanation of the answer is as follows

Q: If B is invertible matrix the show det B = det B that %3D

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Q: IF A is diagonalizable such that A XDX, then the diagonalizing matrix X is unique. %3D Select one: O…

A: Given A is diagonalisable and A = XDX-1

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: Let Mı5(IR) = {A/A and 15th order square matrix with real elements} and consider the linear…

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Q: Let A is an orthogonal matrix, that is, AAT = I and B? = I. Compute the determinanm of the matrix…

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Q: Suppose that A = [c, c2 C3] is a 3 × 3 matrix with columns c;, and suppose that a basis for the…

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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: 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: Let A be an n x n matrix. Then A is nonsingular if and only if its column vectors form a basis for…

A: We have to check

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

A: Solution

Q: If u is a nonzero vector in R", then uu is a rank one matrix. O True O False

A: We have given u is a non-zero vector in ℝn. Then, u=a1a2..an where, at least one  ai≠0 and 1≤i≤n.…

Q: Let A be a known n x n matrix, b be a known n-dimensional vector and z be an unknown vector defined…

A: You asked to solve only (a). (a) Solution:  We will explain here how to compute the determinant of…

Q: If an upper triangular matrix A is singular then at least one diagonal element is zero. O True O…

A: We have to find out

Q: Show that if the non-zero vectors s1), s(2),..., s) are conjugate to a positive definite matrix G,…

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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…

Q: 12. Use the properties of the determinant function to show that if A and B are any n × n matrices,…

A: