3 1 2

3 2 -1

1 1 1

3 2 -1

1 1 1

Work out the determinant: 3(2+1)-1(3+1)+2(3-2)=9-4+2=7.

Take each column and make it into a row:

3 3 1

1 2 1

2 -1 1

Replace each element with its subdeterminant:

(2×1-1×-1)=3 (1×1-1×2)=-1 (1×-1-2×2)=-5

(3×1-(-1)×1)=4 (3×1-1×2)=1 (3×-1-3×2)=-9

(3×1-1×2)=1 (3×1-1×1)=2 (3×2-3×1)=3

Change the sign of every other element and divide by the determinant:

**(** **3 1 -5)**

**(-4 1 9) ÷ 7**

**(1 -2 3)**

Check the result by applying to the original matrix:

( 3 1 -5)(3 1 2)

(-4 1 9)(3 2 -1) ÷ 7

(1 -2 3)(1 1 1)=

(1 0 0)

(0 1 0)

(0 0 1)

The result is the identity matrix, so the inverse is correct.

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