If \( \mathrm{A}=\begin{bmatrix}2 & -3 & 5\\6 & 0 & 4\\1 & 5 & -7\end{bmatrix} \), then which of the following statements are TRUE ?
[where \( \mathrm{A}_{ij} \) is the cofactor of the \( (i,j)^{th} \) element , \( \mathrm{a}_{ij} \) of matrix A, \( 1\ge i,j\le 3 \). ]
(A) \( \mathrm{a}_{11}\,\mathrm{A}_{31}=-24 \)
(B) \( \mathrm{a}_{12}\,\mathrm{A}_{32}=-66 \)
(C) \( \mathrm{a}_{13}\,\mathrm{A}_{33}=-90 \)
(D) \( \mathrm{a}_{21}\,\mathrm{A}_{21}=-24 \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) and (B) only
The cofactor is \( \mathrm{A}_{ij}=(-1)^{i+j}\mathrm{M}_{ij} \), where \( \mathrm{M}_{ij} \) is the minor obtained by deleting row i and column j. The four cofactors needed are:
\( \displaystyle \mathrm{A}_{31}=(+1)\begin{vmatrix}-3 & 5\\0 & 4\end{vmatrix} \)
\( \displaystyle =-12-0=-12 \)
\( \displaystyle \mathrm{A}_{32}=(-1)\begin{vmatrix}2 & 5\\6 & 4\end{vmatrix} \)
\( \displaystyle =-(8-30)=22 \)
\( \displaystyle \mathrm{A}_{33}=(+1)\begin{vmatrix}2 & -3\\6 & 0\end{vmatrix} \)
\( \displaystyle =0+18=18 \)
\( \displaystyle \mathrm{A}_{21}=(-1)\begin{vmatrix}-3 & 5\\5 & -7\end{vmatrix} \)
\( \displaystyle =-(21-25)=4 \)
So \( \mathrm{a}_{11}\mathrm{A}_{31}=2\times(-12)=-24 \), \( \mathrm{a}_{12}\mathrm{A}_{32}=(-3)\times 22=-66 \), \( \mathrm{a}_{13}\mathrm{A}_{33}=5\times 18=90 \) and \( \mathrm{a}_{21}\mathrm{A}_{21}=6\times 4=24 \).
Checking the statements
(A) \( \mathrm{a}_{11}\,\mathrm{A}_{31}=-24 \) — Correct. \( 2\times(-12)=-24 \).
(B) \( \mathrm{a}_{12}\,\mathrm{A}_{32}=-66 \) — Correct. \( (-3)\times 22=-66 \).
(C) \( \mathrm{a}_{13}\,\mathrm{A}_{33}=-90 \) — Incorrect. The value is \( 5\times 18=90 \), not −90.
(D) \( \mathrm{a}_{21}\,\mathrm{A}_{21}=-24 \) — Incorrect. The value is \( 6\times 4=24 \), not −24.
Tip: The sign \( (-1)^{i+j} \) is where most mistakes happen: it is + on the main diagonal and alternates like a chessboard. Forgetting it in (D) would give −24 and make (D) look true.
NCERT (2026-27 edition):
Minors and Cofactors
