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Question
If A = `[("x",0,0),(0,"y",0),(0,0,"z")]` is a non-singular matrix, then find A−1 by using elementary row transformations. Hence, find the inverse of `[(2,0,0),(0,1,0),(0,0,-1)]`
Solution
|A| = `[("x",0,0),(0,"y",0),(0,0,"z")]`
= x(yz) − 0 + 0
= xyz ≠ 0
Since A is a non-singular matrix, A−1 exists.
Consider, AA−1 = I
∴ `[("x",0,0),(0,"y",0),(0,0,"z")] "A"^-1 = [(1,0,0),(0,1,0),(0,0,1)]`
By `(1/"x") "R"_1, (1/"y")"R"_2` and `(1/"z")"R"_3,` we get,
`[(1,0,0),(0,1,0),(0,0,1)] "A"^-1 = [(1/"x",0,0),(0,1/"y",0),(0,0,1/"z")]`
∴ A−1 = `[(1/"x",0,0),(0,1/"y",0),(0,0,1/"z")]`
Comparing `[(2,0,0),(0,1,0),(0,0,-1)]` with `[("x",0,0),(0,"y",0),(0,0,"z")]`
we get, x = 2, y = 1, z = - 1
∴ `1/x = 1/2, 1/y = 1/1 = 1, 1/z = 1/-1 = - 1`
Hence, the inverse of
`[(2,0,0),(0,1,0),(0,0,-1)] "is" [(1/2,0,0),(0,1,0),(0,0,-1)]`
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