Let be matrices, and denote with , respectively  , the transpose of , respectively of .

Then it is necessarily true that:

Let denote the usual matrix multiplication. If 

then

Let denote the usual matrix multiplication. If 

then

Let the following matrices and be given in MATLAB:

A = [3, 1, 2; 4, 1, 0; 3, 1, 4];

B = [6, 2, -3; 2, 4, 1; -3, 1, 13];

Let also the matrices (where is the transpose of ) and be given. 

Which of the following statements is correct?

Consider the square matrix , , where the generic element is given by

Which of the following statements is true?

Let be the matrix given in MATLAB by

M=[1/2,-1/3,1/4,-1/5,1/6;-13/2,11/3,-9/5,7/9,-5/14;10/9,9/8,8/7,7/6,6/5;-1/2,-331/216,-211/252,-127/90,-269/270;-341/18, 463/24, -103/140,751/90,71/14];

Let be the vector space of polynomials with real coefficients of degree . Consider the polynomials , , , , in given by

\begin{align*} p_1(x)=& \frac{1}{2}-\frac{1}{3}x+\frac{1}{4}x^2-\frac{1}{5}x^3+\frac{1}{6}x^4;\\ p_2(x)=&-\frac{13}{2}+\frac{11}{3}x-\frac{9}{5}x^2+\frac{7}{9}x^3-\frac{5}{14}x^4;\\ p_3(x)=&\frac{10}{9}+\frac{9}{8}x+\frac{8}{7}x^2+\frac{7}{6}x^3+\frac{6}{5}x^4;\\ p_4(x)=&-\frac{1}{2}-\frac{331}{216}x-\frac{211}{252}x^2-\frac{127}{90}x^3-\frac{269}{270}x^4;\\ p_5(x)=& -\frac{341}{18}+\frac{463}{24}x-\frac{103}{140}x^2+\frac{751}{90}x^3+\frac{71}{14}x^4. \end{align*}\begin{align*} p_1(x)=& \frac{1}{2}-\frac{1}{3}x+\frac{1}{4}x^2-\frac{1}{5}x^3+\frac{1}{6}x^4;\\ p_2(x)=&-\frac{13}{2}+\frac{11}{3}x-\frac{9}{5}x^2+\frac{7}{9}x^3-\frac{5}{14}x^4;\\ p_3(x)=&\frac{10}{9}+\frac{9}{8}x+\frac{8}{7}x^2+\frac{7}{6}x^3+\frac{6}{5}x^4;\\ p_4(x)=&-\frac{1}{2}-\frac{331}{216}x-\frac{211}{252}x^2-\frac{127}{90}x^3-\frac{269}{270}x^4;\\ p_5(x)=& -\frac{341}{18}+\frac{463}{24}x-\frac{103}{140}x^2+\frac{751}{90}x^3+\frac{71}{14}x^4. \end{align*}

Which of the following statements is correct?

Let be the matrix given in MATLAB by

Let be the matrix given in MATLAB as

Let be a matrix with real entries, diagonalizable over , with an LU factorization (without the need for partial pivoting) and such that . For , let

Consider the square matrix , , where the generic element is given by 

b = [5;12;21;32;45;60;77;96]

Which of the following statements is correct?