Question

Cofactor of 4 in the determinant $\left|\begin{array}{ccc}1& 2& -3\\ 4& 5& 0\\ 2& 0& 1\end{array}\right|$ is equal to

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Answer 1

We can clearly see that element 4 lies in second row and first column which means that here i (row number) = 2 and j (Column number) =1. So the cofactor of this element shall be $(-1{)}^{i+j}{M}_{ij}$ where $M_{ij}$ is the minor of the element. Here i+j = 3 and we know that minor is the determinant of the square matrix formed by deleting the row and column corresponding to the element from the original matrix. Here after deleting the second row and first column we are left with $\left|\begin{array}{cc}2& -3\\ 0& 1\end{array}\right|$ which is equal to $2\times \left(1\right)–(-3)\times 0=2$ which is the minor of the element. So cofactor is $(-1{)}^3\times 2=-2$