Q. 73.8( 33 Votes )

If , find (AB)-1

Answer :


B =


We need to find B-1


To find the inverse of a matrix we need to find the Adjoint of that matrix


For finding the adjoint of the matrix we need to find its cofactors


Let Bij denote the cofactors of Matrix B


Minor of an element bij = Mij �


b11 = 1, Minor of element b11 = M11 = = (3 × 1) – ((-2) × 0) = 3


b12 = 2, Minor of element b12 = M12 = = ((-1) × 1) – (0 × 0) = -1


b13 = -2, Minor of element b13 = M13 = = ((-1) × (-2)) – (3 × 0) = 2


b21 = -1, Minor of element b21 = M21 = = (2 × 1) – ((-2) × (-2)) = -2


b22 = 3, Minor of element b22 = M22 = = (1 × 1) – ((-2) × 0) = 1


b23 = 0, Minor of element b23 = M23 = = (1 × (-2)) – (2 × 0) = -2


b31 = 0, Minor of element b31 = M31 = = (2 × 0) – ((-2) × 3) = 6


b32 = -2, Minor of element b32 = M32 = = (1 × 0) – ((-2) × (-1)) = -2


b33 = 1, Minor of element b33 = M33 = = (1 × 3) – (2 × (-1)) = 5


Cofactor of an element bij, Bij = (-1)i+j × Mij


B11 = (-1)1+1× M11 = 1 × 3 = 3


B12 = (-1)1+2× M12 = (-1) × (-1) = 1


B13 = (-1)1+3× M13 = 1 × 2 = 2


B21 = (-1)2+1× M21 = (-1) × (-2) = 2


B22 = (-1)2+2 × M22 = 1 × 1 = 1


B23 = (-1)2+3 × M23 = (-1) × (-2) = 2


B31 = (-1)3+1 × M31 = 1 × 6 = 6


B32 = (-1)3+2 × M32 = (-1) × (-2) = 2


B33 = (-1)3+3 × M33 = 1 × 5 = 5


Adj B = =


|B| = 1 (3 × 1 – (-2) × 0) – 2 ((-1) × 1 – 0 × 0) + (-2) ((-1) × (-2) – 3 × 0)


|B| = 3 – 2 ( -1 – 0) – 2 (2 – 0)


|B| = 3 + 2 – 4 = 1


B-1 = (Adj B)/|B| = /1 =


We know (AB)-1 = B-1A-1


(AB)-1 = ×


Solving the above matrix we get


(AB)-1 =


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