Questions 5/6 (b) If A and B are matrices in Mag, is A+ B also in Mag? Furthermore, suppose c is areal number; is cA in Mag? (In other words: Is Mag a subspace of Magn?)5. Show that if M is a 3 x 3 magic square with weight w, then we can write M asM = Mo + k.Jwhere Mo is a 3 x 3 magic square with weight 0, J is a 3 x 3 matrix where each entry is 1,and k is a scalar What must k be?6. In this problem, you will find a way to describe all 3 x 3 magic squares. Leta bde fM =9hibe a magic square with weight 0. The conditions on the rows, columns, and diagonals giverise to a homogeneous system containing eight equations and nine unknowns.(a) Write out the system of equations, then solve it (you may use a computer or calculatorto solve the system for you).(b) Show (using a substitution, if necessary) that your solution to the previous part can bewritten in the form-8 - tM =-s +tS - t-ts+t-8(c) Use these results and the result of problem 5 to write an arbitrary 3 x 3 magic squareas a linear combination of three particular linearly independent matrices.7. Find a direct way of showing that the (2, 2)-entry of a 3 x 3 magic square with weight w mustbe w/3.8. Let M be a 3 x 3 magic square of weight 0, obtained from a classical magic square (as inproblem 5). If M has the form given in problem 6(b), write out an equation for the sum ofthe squares of the entries in M. Show that this is the equation of a circle in the variables sand t, and carefully plot it. Show that there are exactly eight points (s, t) on this circle whereboth s and t are integers. Show that these eight points give rise to eight classical 3 x 3 magicsquares. How are these magic squares related to one another?

Given that , A magic square To show that : We can write as Where, is a…

5. Show that if M is a 3 x 3 magic square with weight w, then we can write M as M = Mo + kJ %3D where Mo is a 3 x 3 magic square with weight 0, J is a 3 x 3 matrix where each entry is 1, and k is a scalar What must k be?

5. Show that if M is a 3 x 3 magic square with weight w, then we can write M as M = Mo + kJ %3D where Mo is a 3 x 3 magic square with weight 0, J is a 3 x 3 matrix where each entry is 1, and k is a scalar What must k be?

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter11: Matrices And Determinants

Section11.FOM: Focus On Modeling: Computer Graphics

Problem 6P: Here is a data matrix for a line drawing: D=[012100002440] aDraw the image represented by D. bLet...

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Transformation of Graphs

The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.

Exponential Functions

The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, concepts, structures, graphs, interpreter series, work formulas, and examples of functions involving exponents.

Question

Questions 5/6

(b) If A and B are matrices in Mag, is A+ B also in Mag? Furthermore, suppose c is a
real number; is cA in Mag? (In other words: Is Mag a subspace of Magn?)
5. Show that if M is a 3 x 3 magic square with weight w, then we can write M as
M = Mo + k.J
where Mo is a 3 x 3 magic square with weight 0, J is a 3 x 3 matrix where each entry is 1,
and k is a scalar What must k be?
6. In this problem, you will find a way to describe all 3 x 3 magic squares. Let
a b
de f
M =
9hi
be a magic square with weight 0. The conditions on the rows, columns, and diagonals give
rise to a homogeneous system containing eight equations and nine unknowns.
(a) Write out the system of equations, then solve it (you may use a computer or calculator
to solve the system for you).
(b) Show (using a substitution, if necessary) that your solution to the previous part can be
written in the form
-8 - t
M =
-s +t
S - t
-t
s+t
-8
(c) Use these results and the result of problem 5 to write an arbitrary 3 x 3 magic square
as a linear combination of three particular linearly independent matrices.
7. Find a direct way of showing that the (2, 2)-entry of a 3 x 3 magic square with weight w must
be w/3.
8. Let M be a 3 x 3 magic square of weight 0, obtained from a classical magic square (as in
problem 5). If M has the form given in problem 6(b), write out an equation for the sum of
the squares of the entries in M. Show that this is the equation of a circle in the variables s
and t, and carefully plot it. Show that there are exactly eight points (s, t) on this circle where
both s and t are integers. Show that these eight points give rise to eight classical 3 x 3 magic
squares. How are these magic squares related to one another?

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