consider y=(y1, y2, ... yn) and X= (1 x1, 1 x2, ..., 1 xn). a) show that i) (X^T)X = (n nx, nx Σxi^2). ii) det((X^T)X) = n Sxx. b) state a necessary and sufficient condition on x1, x2, ..., xn for (X^T)X to be invertible. c) show that if (X^T)X is invertible, then (((X^t)X)^-1)(X^T)y = (y-(Sxy/Sxx)x Sxy/Sxx). Hint use the facts that Sxx = Σn, i=1 (xi^2-nx^2), Sxy = Σn,i=1 (xiyi - nxy).

(b) Assume that {v1, v2, · · · , vk} is a basis of Y . Show that the linear transform T : X → Y from (a) is invertible. (Hint: Use (a) to define a suitable linear transform U : Y → X and show that it is the inverse of T .)

Consider the matrix . Which of the following statements is correct?Question 1Select one:Select one:a.None of the other statements are correct.b.The matrix is invertible and its determinant is not equal to zero.c.The matrix is not invertible and its determinant is not equal to zero.d.The matrix is not invertible and its determinant is equal to zero.e.The matrix is invertible and its determinant is equal to zero.

Are the following statements true or false?Answer 1 Question 10 If A is a square matrix with zero determinant, then one row of A must be a sum of two other rows.Answer 2 Question 10 Even if two columns of a square matrix are equal, the determinant of the matrix may be non-zero.Answer 3 Question 10 If any two rows of a square matrix are multiples of each other, then the determinant is always zero

A square matrix A is invertible if and only if

Suppose that A and B are n×n matrices. Check the true statements below: A. If det(A) is zero, then two rows or two columns are the same, or a row or a column is zero.B. det(AT)=(−1)det(A)C. If two row interchanges are made in sucession, then the determinant of the new matrix is equal to the determinant of the original matrix.D. The determinant of A is the product of the diagonal entries in A