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प्रश्न
Find the inverse of the following matrix.
`[(1,2),(2,-1)]`
उत्तर
Let A = `[(1,2),(2,-1)]`
∴ |A| = `|(1,2),(2,-1)|` = − 1 − 4 = − 5 `≠` 0
∴ A−1 exists.
consider AA−1 = I
∴ `[(1,2),(2,-1)]`A−1 = `[(1,0),(0,1)]`
By R2 − 2R1, we get
`[(1,2),(0,-5)]`A−1 = `[(1,0),(-2,1)]`
By `(-1/5)`R2, we get,
`[(1,2),(0,1)]`A−1 = `[(1,0),(2//5,-1//5)]`
By R1 − 2R2, we get,
`[(1,0),(0,1)]`A−1 = `[(1//5,2//5),(2//5,-1//5)]`
∴ A−1 = `-1/5[(-1,-2),(-2,1)]`
∴ A−1 = `1/5[(1,2),(2,-1)]`
The answer can be checked by finding the product AA−1.
AA−1 = `[(1,2),(2,-1)][(1//5,2//5),(2//5,-1//5)]`
= `[(1(1/5) + 2(2/5),1(2/5) + 2(-1/5)),(2(1/5) - 1(2/5),2(2/5) -1(-1/5))]`
= `[(1/5 + 4/5,2/5 - 2/5),(2/5 - 2/5,4/5 + 1/5)] = [(1,0),(0,1)]` = I
Hence, A−1 is the required answer.
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