Question

Write Minors and Cofactors of the elements of following determinants: 
 I. \(\begin{vmatrix}2&-4\\0&3\end{vmatrix}\) 

 II. \(\begin{vmatrix}a&c\\b&d\end{vmatrix}\)

Answer 1

I. The given determinant is \(\begin{vmatrix}2&-4\\0&3\end{vmatrix}\)
Minor of element a_ij is M_ij.
∴M₁₁ = minor of element a₁₁ = 3
M₁₂ = minor of element a₁₂ = 0
M₂₁ = minor of element a₂₁ = −4
M₂₂ = minor of element a₂₂ = 2
Cofactor of a_ij is A_ij = (−1)^i+j M_ij. 
∴A₁₁ = (−1)¹⁺¹ M₁₁ = (−1)² (3) = 3 
A₁₂ = (−1)¹⁺² M₁₂ = (−1)³
(0) = 0
A₂₁ = (−1)²⁺¹ M₂₁ = (−1)³
(−4) = 4
A₂₂ = (−1)²⁺² M₂₂ = (−1)4
(2) = 2

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(ii) The given determinant is \(\begin{vmatrix}a&c\\b&d\end{vmatrix}\)
Minor of element a_ij is M_ij.
∴M₁₁ = minor of element a₁₁ = d
M₁₂ = minor of element a₁₂ = b
M₂₁ = minor of element a₂₁ = c
M₂₂ = minor of element a₂₂ = a
Cofactor of a_ij is A_ij = (−1)^i+j M_ij
∴A₁₁ = (−1)¹⁺¹ M¹¹ = (−1)²
(d) = d
A₁₂ = (−1)¹⁺² M₁₂ = (−1)³
(b) = −b
A₂₁ = (−1)²⁺¹ M₂₁ = (−1)³
(c) = −c
A₂₂ = (−1)²⁺² M₂₂ = (−1)⁴
(a) = a