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Engineering

Mathematics

Introduction to Determinants

Question

$Adj [\begin{array}{ccc} 1 & 0 & 2 \\ - 1 & 1 & - 2 \\ 0 & 2 & 1 \end{array}] = [\begin{array}{ccc} 5 & a & - 2 \\ 1 & 1 & 0 \\ - 2 & - 2 & b \end{array}] \Rightarrow [\begin{matrix} a & b \end{matrix}] =$ ____

[ – 4 1]

[ – 4 – 1]

[4 1]

[4 – 1]

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Given,

Adj [\begin{array}{ccc} 1 & 0 & 2 \\ - 1 & 1 & - 2 \\ 0 & 2 & 1 \end{array}] = [\begin{array}{ccc} 5 & 9 & - 2 \\ 1 & 1 & 0 \\ - 2 & - 2 & 6 \end{array}]

Adjoint of matrix = Transpose of cofactor matrix

C_{11} = |\begin{array}{cc} 1 & - 2 \\ 2 & 1 \end{array}| = 5

C_{12} = |\begin{array}{cc} - 1 & 1 \\ 0 & 2 \end{array}| = - 2

C_{13} = |\begin{array}{cc} - 1 & 1 \\ 0 & 2 \end{array}| = - 2

C_{21} = - |\begin{array}{cc} 0 & 2 \\ 2 & 1 \end{array}| = 4

C_{22} = |\begin{array}{cc} 1 & 2 \\ 0 & 1 \end{array}| = 1

C_{23} = - |\begin{array}{cc} 1 & 0 \\ 0 & 2 \end{array}| = - 2

C_{31} = |\begin{array}{cc} 0 & 2 \\ 2 & 1 \end{array}| = - 4

C_{32} = - |\begin{array}{cc} 1 & 2 \\ - 1 & - 2 \end{array}| = 0

C_{33} = |\begin{array}{cc} 1 & 0 \\ - 1 & 1 \end{array}| = 1

Cofactor matrix

C = [\begin{array}{ccc} 5 & 1 & - 2 \\ 4 & 1 & - 2 \\ - 4 & 0 & 1 \end{array}]

Adj matrix

= [\begin{array}{ccc} 5 & 4 & - 4 \\ 4 & 1 & 0 \\ - 2 & - 2 & 1 \end{array}] = [\begin{array}{ccc} 5 & a & - 2 \\ 1 & 1 & 0 \\ - 2 & - 2 & b \end{array}]

[a b] = [4 1]

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