Question

Let Aequals=left bracket Start 2 By 2 Matrix 1st Row 1st Column 1 2nd Column 2...

Let Aequals=left bracket Start 2 By 2 Matrix 1st Row 1st Column 1 2nd Column 2 2nd Row 1st Column 8 2nd Column 18 EndMatrix right bracket

1 2
8 18

​, Bold b 1b1equals=left bracket Start 2 By 1 Matrix 1st Row 1st Column negative 5 2nd Row 1st Column negative 36 EndMatrix right bracket

−5
−36

​, Bold b 2b2equals=left bracket Start 2 By 1 Matrix 1st Row 1st Column 3 2nd Row 1st Column 16 EndMatrix right bracket

3
16

​, Bold b 3b3equals=left bracket Start 2 By 1 Matrix 1st Row 1st Column 3 2nd Row 1st Column 22 EndMatrix right bracket

3
22

​, and Bold b 4b4equals=left bracket Start 2 By 1 Matrix 1st Row 1st Column 4 2nd Row 1st Column 24 EndMatrix right bracket

4
24

.​(a) Find

Upper A Superscript negative 1A−1

and use it solve the four equations

Axequals=Bold b 1b1​,

Axequals=Bold b 2b2

​,Axequals=Bold b 3b3​,

and

Axequals=Bold b 4b4.

​(b) The four equations in part​ (a) can be solved by the same set of​ operations, since the coefficient matrix is the same in each case. Solve the four equations in part​ (a) by row reducing the augmented matrix​ [A

Bold b 1b1

Bold b 2b2

Bold b 3b3

Bold b 4b4​].

Find

Upper A Superscript negative 1A−1.

Select the correct choice below​ and, if​ necessary, fill in the answer boxes to complete your choice

A.Upper A Superscript negative 1A−1equals=left bracket Start 2 By 2 Matrix 1st Row 1st Column nothing 2nd Column nothing 2nd Row 1st Column nothing 2nd Column nothing EndMatrix right bracket

9 −1
−4 12

​(Simplify your​ answers.)

B.

The matrix is not invertible.

Solve

Axequals=Bold b 1b1.

xequals=left bracket Start 2 By 1 Matrix 1st Row 1st Column nothing 2nd Row 1st Column nothing EndMatrix right bracket

−9
2

​(Simplify your​ answers.)Solve

Axequals=Bold b 2b2.

xequals=left bracket Start 2 By 1 Matrix 1st Row 1st Column nothing 2nd Row 1st Column nothing EndMatrix right bracket

11
−4

​(Simplify your​ answers.)Solve

Axequals=Bold b 3b3.

xequals=left bracket Start 2 By 1 Matrix 1st Row 1st Column nothing 2nd Row 1st Column nothing EndMatrix right bracket

5
−1

​(Simplify your​ answers.)Solve

Axequals=Bold b 4b4.

xequals=left bracket Start 2 By 1 Matrix 1st Row 1st Column nothing 2nd Row 1st Column nothing EndMatrix right bracket

12
−4

​(Simplify your​ answers.)​(b) Solve the four equations by row reducing the augmented matrix​ [A

Bold b 1b1

Bold b 2b2

Bold b 3b3

Bold b 4b4​].

Write the augmented matrix ​[A

Bold b 1b1

Bold b 2b2

Bold b 3b3

Bold b 4b4​]

in reduced echelon form.left bracket Start 2 By 6 Matrix 1st Row 1st Column nothing 2nd Column nothing 3rd Column nothing 4st Column nothing 5st Column nothing 6st Column nothing 2nd Row 1st Column nothing 2nd Column nothing 3rd Column nothing 4st Column nothing 5st Column nothing 6st Column nothing EndMatrix right bracket nbsp

​(Simplify your​ answers.)

Are the solutions the same in​ (a) and​ (b)?

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