Question

# The adjoint of the matrix ⎡⎢⎣123111234⎤⎥⎦ is

A
121121121
B
111222111
C
112213314
D
211211211

Solution

## The correct option is B ⎡⎢⎣11−1−2−2211−1⎤⎥⎦Calculation of adjoint is a step by step process. You need to know minor and cofactor of an element first. We first find the cofactors of all elements to create a cofactor matrix. Simply taking the transpose of it will give the adjoint matrix From the video you saw that minor of an element aij is given by the determinant of matrix formed by excluding the row and column of that particular element. α12 If minor of an element aij is Mij, then its Cofactor Aij  is given by (−1)2 Aij = (−1)i+j Mij Lets take each element of the give matrix and find their cofactors. Cofactor of α11=(−1)2∣∣∣1134∣∣∣=1.(4−3)=1Cofactor of α12=(−1)3∣∣∣1124∣∣∣=(−1)(2)=−2 Cofactor of α13=(−1)4∣∣∣1123∣∣∣=1Cofactor of α21=(−1)3∣∣∣2334∣∣∣=(−1)(−1)=+1 Cofactor of α22=(−1)4∣∣∣1324∣∣∣=−2Cofactor of α23=(−1)5∣∣∣1223∣∣∣=(−1)(−1)=1 Cofactor of α31=(−1)4∣∣∣2311∣∣∣=(−1) Cofactor of α32=(−1)5∣∣∣1311∣∣∣=(−1)(−2)=2 Cofactor of α33=(−1)6∣∣∣1211∣∣∣=1−1=−1 Cofactor matrix = ⎡⎢⎣1−211−21−12−1⎤⎥⎦ Now take the transpose of cofactor matrix to get Adjoint of the Matrix. ∴Adj(A)=⎡⎢⎣11−1−2−2211−1⎤⎥⎦ Mathematics

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