2.2 Q-10

Exercise 2.2 — Question 10

This question involves evaluating a $3\times 3$ determinant using cofactor expansion and properties of determinants.

Key Concepts

Cofactor Expansion (along Row 1):

For a $3\times 3$ matrix $A$:

$A=\left(\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right)$

The determinant is:

$|A|=a_{11}{C}_{11}+a_{12}{C}_{12}+a_{13}{C}_{13}$

where the cofactor $C_{ij}=(-1{)}^{i+j}{M}_{ij}$ and $M_{ij}$ is the minor obtained by deleting row $i$ and column $j$.

Sign Pattern for Cofactors:

$\left(\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}\right)$

Useful Properties of Determinants

1. If two rows (or columns) are identical, $|A|=0$.
2. Interchanging two rows changes the sign of the determinant.
3. Multiplying a row by a scalar $k$ multiplies the determinant by $k$.
4. Adding a multiple of one row to another does not change the determinant.

Method

1. Choose a row or column for expansion (prefer one with zeros).
2. For each element in that row/column, compute its cofactor.
3. Multiply each element by its cofactor and sum the results.