Create a non-homogeneous system of five linear equations in five unknowns Az = 6, satisfying the following conditions. Here, the matrix A = a11 a21 a31 a41 a51 a15 a12 a13 a14 a22 23 24 25 a32 a33 a34 a35 a42 a43 a44 a45 a52 a53 a54 a55 should be such that no aij = 0 and no aj repeats; that is ai; = alk if and only if i = 1 and j = k. Furthermore, the solution vector

Determine a least square solution to the matrix equation A = b ,where 1 2 6 A 3 3 and 6 3 0 0

Given the matrix A = 0 2 2 0 2 2 find an orthogonal diagonalization of A: A = QDQ" Enter each matrix Q and D in the format [[a11 , a12 , a13] , [a21 , a22 , a23] , [a31, a32, a33]]. Note: if you need to enter a square root of a number x you can do it as follows: sqrt (x). Alternatively, you can compute matrix entries in the decimal form rounded to 3 decimal places, and enter them in this. Matrix Q: Matrix D:

Let A be a 3x3 matrix, and let u₁ and u₂ be the first two columns of A. Assume that u₁ and u₂ are not parallel. If a linear system Av = b has infinitely many solutions, what can we say about the third column u3 of A, in terms of u₁ and u₂? In other words, how does u3 relate to u₁ and u₂?