Let A be a nilpotent matrix . Prove that there is an orthogonal matrix Q such that QT AQ is upper triangular with zeros on its diagonal.

Q: Let A be an n × n matrix with triangular factorization LU. Show that det(A) = u11u22 · · · unn

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Q: Prove that the nonzero row vectors of a matrix in row-echelon form are linearly independent. Let a =…

A: Let A=aij be an m×n matrix in row echelon form.  Given that first column of A is not all zero, and…

Q: Prove that the column vectors of the matrix B = space of the matrix B. U 2 128 0 4 2 are dependent.…

A: The given matrix .The aim is to prove that the column vectors of  are dependent and describe the…

Q: Prove that the n × n identity matrix is orthogonal.

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Q: Prove that A2 # 0. Let A € M4×4(F) be a diagonalizable matrix with rank 1.

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Q: Construct a 4 x 3 matrix A = [ā₁ ä2 ã3] such that {a1, a2} is a linearly dependent set and a3 is not…

A: two vectors and spanning set of two vectors. .So take such vectors as required are in the following…

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: Prove that every unit lower triangular matrix is invertible and that its inverse is also unit lower…

A: It has to be proven that every unit lower triangular matrix is invertible. And it is also be proven…

Q: Determine the Householder matrix that annihilates all but the first entry of the vector v = [2 22…

A: Let α=2222T Then take, v=α-αc1=2222-α2000=2222-α000 ,Here α=±||a||2=±2 Now , since,α1  is positive ,…

Q: 3) Let A = 2 -1 Find a matrix P that orthogonally diagonalizes A. Use P to com- 0 -1 pute A5.

A: Given, A=11012-10-11first we have to find a matrix Pfor eigen values of…

Q: Let A be an orthogonal matrix. Show that cA is orthogo- nal if and only if c = ±1.

A: Given:            A be an orthogonal matrix.

Q: Construct a 3 x 3 matrix A with nonzero entries, and a vector b in R³ such that b is not in the set…

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Q: Let A and C be n×n matrices such that ​CA=I​ (the n×n identity​ matrix). Show that Ax​=0 has only…

A: Here, CA=I, where I is the identity matrix.

Q: xplain why A can be changed into I using only row replacements and scaling. (Where are the pivots?)…

A: We have to fill the empty spaces in the given proof.

Q: Find a matrix P such that P'AP orthogonally diagonalizes A. Verify that PTAP gives the proper…

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Q: Show that if A is an m × n matrix such that AAT is non- singular, then rank A = m.

A: Let AAT can be non singular only if the columns in A are linear independent and they span the column…

Q: Let A be an n × n matrix with a column of zeros. Show directly from the Leibniz formula that…

A: Let, A be an n×n matrix with a column of zeros.  To prove detA=0 using Leibnitz formula for…