2.3 Q-1

Exercise 2.3 — Q1: Evaluating 3×3 Determinants Using Cofactors

Key Definitions

Minor $M_{ij}$: The determinant of the $2\times 2$ submatrix obtained by deleting row $i$ and column $j$ from a $3\times 3$ matrix.

Cofactor $C_{ij}$: The signed minor defined by $C_{ij}=(-1{)}^{i+j}{M}_{ij}$

Sign Pattern for a $3\times 3$ Matrix: $\left(\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}\right)$

Cofactor Expansion (Along Row 1)

For a $3\times 3$ matrix $A=\left(\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right)$, the determinant expanded along the first row is:

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

$=a_{11}\left|\begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right|-a_{12}\left|\begin{array}{cc}{a}_{21}& a_{23}\\ a_{31}& a_{33}\end{array}\right|+a_{13}\left|\begin{array}{cc}{a}_{21}& a_{22}\\ a_{31}& a_{32}\end{array}\right|$

Singular vs Non-Singular Matrices

Worked Example

Evaluate $\left|\begin{array}{ccc}1& 2& 3\\ 0& 4& 5\\ 1& 0& 6\end{array}\right|$ by cofactor expansion along Row 1.

Step 1: Identify $a_{11}=1,a_{12}=2,a_{13}=3$.

Step 2: Compute each cofactor: $C_{11}=(+1)\left|\begin{array}{cc}4& 5\\ 0& 6\end{array}\right|=(4)(6)-(5)(0)=24$ $C_{12}=(-1)\left|\begin{array}{cc}0& 5\\ 1& 6\end{array}\right|=-(0\cdot 6-5\cdot 1)=-(-5)=5$ $C_{13}=(+1)\left|\begin{array}{cc}0& 4\\ 1& 0\end{array}\right|=(0)(0)-(4)(1)=-4$

Step 3: Expand: $|A|=1(24)+2(5)+3(-4)=24+10-12=22$

Properties of Determinants (Key)

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 scalar $k$ multiplies the determinant by $k$.
4. Adding a multiple of one row to another does not change the determinant.