NCERT Class 12 Maths · Chapter 4 Determinants · Exercise 4.3 Question 2

Question

Write the minors and cofactors of every element of (i) $\left[\begin{matrix}1 & 0 & 0\0 & 1 & 0\0 & 0 & 1\end{matrix}\right]$ and (ii) $\left[\begin{matrix}1 & 0 & 4\3 & 5 & -1\0 & 1 & 2\end{matrix}\right]$.

Accepted Answer

Final Answer: (1) minors $\left[\begin{matrix}1 & 0 & 0\0 & 1 & 0\0 & 0 & 1\end{matrix}\right]$, cofactors $\left[\begin{matrix}1 & 0 & 0\0 & 1 & 0\0 & 0 & 1\end{matrix}\right]$; (2) minors $\left[\begin{matrix}11 & 6 & 3\-4 & 2 & 1\-20 & -13 & 5\end{matrix}\right]$, cofactors $\left[\begin{matrix}11 & -6 & 3\4 & 2 & -1\-20 & 13 & 5\end{matrix}\right]$ Step 1: For each Mᵢⱼ delete row i and column j and evaluate the remaining determinant. A cofactor is Cᵢⱼ=(−1)ⁱ⁺ʲMᵢⱼ. Step 2: Part 1: the complete minor matrix is $\left[\begin{matrix}1 & 0 & 0\0 & 1 & 0\0 & 0 & 1\end{matrix}\right]$. Step 3: Apply the checkerboard signs (+,−,+; −,+,−; +,−,+), truncated to the matrix order: the cofactor matrix is $\left[\begin{matrix}1 & 0 & 0\0 & 1 & 0\0 & 0 & 1\end{matrix}\right]$. Step 4: Part 2: the complete minor matrix is $\left[\begin{matrix}11 & 6 & 3\-4 & 2 & 1\-20 & -13 & 5\end{matrix}\right]$. Step 5: Apply the checkerboard signs (+,−,+; −,+,−; +,−,+), truncated to the matrix order: the cofactor matrix is $\left[\begin{matrix}11 & -6 & 3\4 & 2 & -1\-20 & 13 & 5\end{matrix}\right]$.

NCERT Class 12 Maths · Chapter 4 Determinants · Exercise 4.3 Question 2

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