Solution Of System Of Equations Using Matrix Method: A Step-By-Step Approach

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Solution of System of Equations Using Matrix Method: A Step - by - Step Approach Question Solve the following system of equations using matrix method: 𝟐 𝒙 + 𝟑 𝒚 + 𝟏𝟎 𝒛 = 𝟒 , 𝟒 𝒙 − 𝟔 𝒚 + 𝟓 𝒛 = 𝟏 , 𝟔 𝒙 + 𝟗 𝒚 − 𝟐𝟎 𝒛 = 𝟐 Topic: Matrix Sub Topic: Solution of system of equations by matrix method CLASS - XII Solution: Step - 1 – Let us assume that: 1 𝑥 = 𝑎 , 1 𝑦 = 𝑏 and 1 𝑧 = 𝑐 , the system takes the form: 2 𝑎 + 3 𝑏 + 10 𝑐 = 4 ………………… (1) 4 𝑎 − 6 𝑏 + 5 𝑐 = 1 …………………… (2) 6 𝑎 + 9 𝑏 − 20 𝑐 = 2 …………………. (3) Step - 2 - Matrix equation and notations The above system is given by the matrix equation: AX = B OR X = 𝑨 − 𝟏 𝑩 Where A = [ 2 3 10 4 − 6 5 6 9 − 20 ] 3 𝑋 3 , B = [ 4 1 2 ] 3 𝑋 1 , X = [ 𝑎 𝑏 𝑐 ] 3 𝑋 1 Step - 3 - Determinant of A | 𝐴 | = | 2 3 10 4 − 6 5 6 9 − 20 | = 1200 ≠ 0 ⇒ System has a unique solution and hence consistent. Step - 4 - Cofactors 𝐴 11 = 75 , 𝐴 12 = 110 , 𝐴 13 = 72 𝐴 21 = 150 , 𝐴 22 = − 100 , 𝐴 23 = 0 𝐴 31 = 75 , 𝐴 32 = 30 , 𝐴 33 = − 24 Step - 5 - Adjoint of A
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