2.2 Q-9

Exercise 2.2 — Question 9

This question requires evaluating a $3\times 3$ determinant using cofactor expansion and/or 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.
5. If a row (or column) consists entirely of zeros, $|A|=0$.

Worked Example

Evaluate:

$\left|\begin{array}{ccc}2& 1& 3\\ 0& -1& 2\\ 4& 0& 1\end{array}\right|$

Expanding along Row 1:

$=2\cdot (-1{)}^{1+1}\left|\begin{array}{cc}-1& 2\\ 0& 1\end{array}\right|-1\cdot (-1{)}^{1+2}\left|\begin{array}{cc}0& 2\\ 4& 1\end{array}\right|+3\cdot (-1{)}^{1+3}\left|\begin{array}{cc}0& -1\\ 4& 0\end{array}\right|$

$=2((-1)(1)-(2)(0))+1((0)(1)-(2)(4))+3((0)(0)-(-1)(4))$

$=2(-1)+1(-8)+3(4)$

$=-2-8+12=2$