Write Minors and Cofactors of the elements of following determinants:
I. \(\begin{vmatrix}2&-4\\0&3\end{vmatrix}\)
II. \(\begin{vmatrix}a&c\\b&d\end{vmatrix}\)
Write Minors and Cofactors of the elements of following determinants:
I. \(\begin{vmatrix}2&-4\\0&3\end{vmatrix}\)
II. \(\begin{vmatrix}a&c\\b&d\end{vmatrix}\)
Solution and Explanation
I. The given determinant is \(\begin{vmatrix}2&-4\\0&3\end{vmatrix}\)
Minor of element aij is Mij.
∴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 aij is Aij = (−1)i+j Mij.
∴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
(ii) The given determinant is \(\begin{vmatrix}a&c\\b&d\end{vmatrix}\)
Minor of element aij is Mij.
∴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 aij is Aij = (−1)i+j Mij
∴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
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