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# Using Elementary Transformation, Find the Inverse of the Matrix : [(0,1,2),(1,2,3),(3,1,1)] - ISC (Science) Class 12 - Mathematics

ConceptMatrices Proof of the Uniqueness of Inverse

#### Question

Using elementary transformation, find the inverse of the matrix :

[(0,1,2),(1,2,3),(3,1,1)]

#### Solution

Let A = [(0,1,2),(1,2,3),(3,1,1)]

|A| = 0[2-3]-1[1-9] + 2[1-6]

= 8 + 2(-5) = 8 - 10 = -2 != 0

=> A^(-1) exists

:. A = IA

[(0,1,2),(1,2,3),(3,1,1)] = [(1,0,0),(0,1,0),(0,0,1)]A

R_1 ↔ R_2

[(0,1,2),(1,2,3),(3,1,1)] = [(1,0,0),(0,1,0),(0,0,1)]A

R_3"/"R_3 - 3R_3

[(1,2,3),(0,1,2),(0,-5,-8)] = [(0,1,0),(1,0,0),(0,-3,1)]A

R_3"/"R_3 + 5R_2

[(1,2,3),(0,1,2),(0,0,2)] = [(0,1,0),(1,0,0),(5, -3, 1)]A

R_2 "/"R_2 - R_3

[(1,2,3),(0,1,0),(0,0,2)] = [(0,1,0),(-4,3,-1),(5,-3,1)]A

R_1"/"R_1 -2R_2 and R_3 xx  1/2

[(1,0,3),(0,1,0),(0,0,1)] = [(8,-5,2),(-4,3,-1),(5/3, (-3)/2, 1/2)]A

R_1"/"R_1 - 3R_3

[(1,0,0),(0,1,0),(0,0,1)] = [(1/2, -1/2 , 1/2),(-4, 3,-1),(5/2, (-3)/2 , 1/2)]A

:. A^(-1) = 1/2[(1,-1,1),(-8, 6, -2), (5,-3,1)]

Is there an error in this question or solution?

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Solution Using Elementary Transformation, Find the Inverse of the Matrix : [(0,1,2),(1,2,3),(3,1,1)] Concept: Matrices - Proof of the Uniqueness of Inverse.
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