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Question
Find the inverse of the matrix `[ (1, 2, 3), (1, 1, 5), (2, 4, 7)]` by using the adjoint method.
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Solution
Let A = `[ (1, 2, 3), (1, 1, 5), (2, 4, 7)]`
`A_11 = (-1)^(1+1) |(1, 5), (4,7)| = -13`
`A_12 = (-1)^(1+2) |(1, 5), (2,7)| = 3`
`A_13 = (-1)^(1+3) |(1, 1), (2,4)| = 2`
`A_21 = (-1)^(2+1) |(2, 3), (4,7)| = -2`
`A_22 = (-1)^(2+2) |(1, 3), (2,7)| = 1`
`A_23 = (-1)^(2+3) |(1, 2), (2,4)| = 0`
`A_31 = (-1)^(3+1) |(2, 3), (1,5)| = 7`
`A_32 = (-1)^(3+2) |(1, 3), (1,5)| = -2`
`A_33 = (-1)^(3+3) |(1, 2), (1,1)| = -1`
`|A| = |(1, 2, 3), (1, 1, 5), (2, 4, 7)|`
= 1( 7 - 20 ) - 2( 7 - 10 ) + 3( 4 - 2 )
= 1( -13 ) -2( -3 ) + 3( 2 )
= -13 + 6 +6
= -1
Adj. (A) = `|(-13, -2, 7), (3, 1, -2), (2, 0, -1)|`
`A^-1 = [Adj. A]/|A| = 1/|A| (Adj. A)`
= `1/-1 |(-13, -2, 7), (3, 1, -2), (2, 0, -1)|`
`A^-1 = 1/-1 |(13, 2, -7), (-3, -1, 2), (-2, 0, 1)|`
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