Matrices & Determinants Question - None

Question

If Δ =

|\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}|

and A1, B1, C1 denote the co-factors of a1, b1, c1 respectively, then the value of the determinant

|\begin{matrix} A_{1} & B_{1} & C_{1} \\ A_{2} & B_{2} & C_{2} \\ A_{3} & B_{3} & C_{3} \end{matrix}|

is -

Accepted Answer

Let A =

|\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}|

= \Delta. The matrix of cofactors is C =

|\begin{matrix} A_{1} & B_{1} & C_{1} \\ A_{2} & B_{2} & C_{2} \\ A_{3} & B_{3} & C_{3} \end{matrix}|

. We know that the adjoint of a matrix A, adj(A), is the transpose of the cofactor matrix. Thus, |adj(A)| = |C|^T = |C|. Also, |adj(A)| = |A|^n-1, where n is the order of the matrix. Here n=3, so |adj(A)| = |A|^3-1 = |A|² = \Delta².

Question:

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