Solve the following three equations for x 1 , x 2 , x 3 in terms of b 1 , b 2 ,

Question

Solve the following three equations for x 1 , x 2 , x 3 in terms of b 1 , b 2 , b 3 . ( Hint : Use matrix multiplication to write-out each of the three equations.) (2 pts)

[1 0 0 [x1    [ b 1

1 1 0     x2   =     b2

1 1 1]    x3]         b3]

(b) The equation above can be written in “shorthand” form A x = b . Use the results of part (a) to write the solution in the form x = A 1 b and determine A 1 . (You must show your work—you can use MATLAB only to check your answer.) (2 pts)

(c) Are the columns of A dependent or independent ? ( circle one ) (1 pt)

Explanation / Answer

(b)Let x=[x1

x2

x3]

Ax=b

[ 1 0 0 [x1 [b1

1 1 0 x2 = b2

1 1 1] x3] b3]

Now, if the above A has an inverse, then both sides can be left-multiplied by A1 to get

Ax=b

A-1 Ax=A-1 b

I2 x=A-1 b

x=A-1 b

=[1 0 0 [b1

-1 1 0 b2 =[1*b1+0+0 [b1

0 -1 1] b3] -1*b1+1*b2+0 = -b1+b2

0-1*b2+1*b3] -b2+b3]

(c)The columns of A are independent.

In matlab

Declare the system of equations.

Use equationsToMatrix to convert the equations into the form AX = B. The second input to equationsToMatrix specifies the independent variables in the equations.

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