Suppose A is invertible and orthogonally diagonalizable. Explain why A–1 is also orthogonally diagonalizable.

Q: If A is diagonalizable and B is diagonalizable, must it be true that AB is diagonalizable? Give a…

A: If A is diagonalizable and B is diagonalizable then it is not necessary for AB to be diagonalizable.…

Q: Prove that A is invertible if and only if A = QR, where Q is orthogonal and R is upper triangular…

A: We need to prove that, if A is invertible if and only if A=QR where Q is orthogonal and R is upper…

Q: If you are given a matrix M such that M = [p 4 5 q 5 4  r 0 5] , define the vector [p q r]…

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Q: 1 1 -2) 13. Find the rank of A=| 2 1 |and explain why it indicates that A is (-1 -1 2 not…

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Q: Find the conjugate of z = u+w If u = 6+2i,z= 1+4i

A: Step 1: The conjugate of a complex number z=a+bi is: zˉ=a−bi i.e., it is obtained by changing the…

Q: Prove that if A and B are orthogonally diagonalizable then so is A + B.

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Q: Consider the 2 X 2 matrices A1 = 1 1 A2 = 0 1 A3 = 0 0 A4 = 1 0 1 1 1 0 () Argue whether these four…

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Q: Find two examples to demonstrate that if (a-d)2+4bc =0 then A may or may be diagonalizable

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Q: a1 For a fixed vector a = a2 E R³, we define the cross product as аз |a2x3 azx2 ахх%— a3X1 – a1x3…

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Q: If L and U are invertible, then (LU) – Find A from the given LU factorization: 9 -27 7-14 96 = -63…

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Q: Consider the 2 X 2 matrices A1 =  1      1           A2 =  0      1        A3  =   0      0      A4…

A: Thus, the four matrices are not linearly independent.

Q: Let S be a set of n vectors of dimension m with m>n and let A be the m x n matrix whose columns are…

A: Given: A set S of vectors of dimension m with m>n and Am×n a matrix whose columns are vectors in…

Q: Show that if a matrix U is in row echelon form, then the nonzero row vectors of U form a basis for…

A: It is given that a matrix U that is in row echelon form. So, the nonzero row of that matrix are…

Q: Find the 2 x 2 matrix P such that Pu is the orthogonal projection of in R² onto the line consisting…

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Q: Show M={A in M2,2| |A| =0} is not a vector space under the usual addition and scalar multiplication…

A: We will show M={A in M2,2| |A| =0} Is not vector space by exhibiting two elements A,B belongs M but…

Q: Let 1 2 A = 3 6 Is A invertible? Explain your answer.

A: To verify is the given matrix A is invertible Or not.

Q: If A is an n x n matrix and b + 0 in R", then consider the set of solutions to Ax = b. Select true…

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Q: Use the least squares method to find the orthogonal projection of b = [1 -2 2] onto the column space…

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Q: Let 1-1-41 1-52 2 A = 6= 1 3 1 0 -1 -3 -1 a. Find the orthogonal projection of b onto Col(A). proj…

A: Step 1: Step 2: Step 3: Step 4:

Q: True/False: If A is diagonalizable, then so is AT.

A: A matrix is said to be diagonalizable if a matrix of n×n elements has n linear independent eigen…

Q: Show that if A is a mxn matrix and rank(A)=m then Ax=b has a solution for any vector b in Fm (Hint:…

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Q: a) Explain what it means, by definition, that a polynomial m(x) € P(R) is the minimal poly- nomial…

A: Definition Let  be a A*A matrix. An annihilating polynomial  (i.e., such that m(x)=0) is called a…

Q: Let p(x) = a + a₁x + a₂x² and g(x) = bo + b₁x + b₂x² be vectors in P₂ with (p, q) = aobo + a1b₁ +…

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Q: What is the matrix of the transpose map M2×2(F) → M2×2(F) with respect to the "standard basis" of…

A: Given linear transformation is T(A) = AT

Q: Find a basis of the following subspace of R³: W linear independence. {(a+b,b+c, c-a)}. Hint:…

A: Given Subspace is, W={(a+b,b+c,c-a):a,b,c∈ℝ}   Now, (a+b,b+c,c-a) =a(1,0,-1)+b(1,1,0)+c(0,1,1)   So,…

Q: Let y=[1,6] and u=[5,7] write y as the sum of two orthogonal vectors , one in span {u} and one…

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Q: If both matrices A,B are orthogonal prove that (ab) is also orthogonal

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Q: Let A,B ∈ Matn(F ) denote two square matrices. Assume that A and B are row equivalent. Show that B…

A: The objective of this question is to prove that if two square matrices A and B are row equivalent,…

Q: Find an explicit description of the null space of matrix A by listing vectors that span the null…

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Q: Prove that the matrices and a a a 0. a 3. a are Similar ( These 2 matrices are not deagonalısable)

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Q: Let T(x) = Ax for the matrix A and [4 -4 -16] -16, b = A = 7 5 Lo 0 1 1 Select true or false for…

A: Given that  A=4-4-1675-16011, b=3-11, c=-451 Select true or false statement.