2.3 Determinants and Cofactor Expansion

Minor of an Element

For a $3\times 3$ matrix $A$, the minor $M_{ij}$ of element $a_{ij}$ is the determinant of the $2\times 2$ submatrix obtained by deleting row $i$ and column $j$.

Example: For 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 minor of $a_{11}$ is: $M_{11}=\left|\begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right|=a_{22}{a}_{33}-a_{23}{a}_{32}$

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Cofactor of an Element

The cofactor $A_{ij}$ of element $a_{ij}$ is defined as: $A_{ij}=(-1{)}^{i+j}{M}_{ij}$

The sign pattern $(-1{)}^{i+j}$ for a $3\times 3$ matrix is: $\left(\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}\right)$

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Cofactor Expansion (Laplace Expansion)

The determinant of a $3\times 3$ matrix can be evaluated by expanding along any row or column.

Expansion along Row 1:

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

Substituting the cofactor formula: $|A|=a_{11}{M}_{11}-a_{12}{M}_{12}+a_{13}{M}_{13}$

Worked Example

Evaluate $|A|$ for: $A=\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right)$

Expanding along Row 1:

$M_{11}=\left|\begin{array}{cc}5& 6\\ 8& 9\end{array}\right|=45-48=-3$

$M_{12}=\left|\begin{array}{cc}4& 6\\ 7& 9\end{array}\right|=36-42=-6$

$M_{13}=\left|\begin{array}{cc}4& 5\\ 7& 8\end{array}\right|=32-35=-3$

$|A|=1(+1)(-3)+2(-1)(-6)+3(+1)(-3)=-3+12-9=0$

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Properties of Determinants

These properties simplify determinant evaluation:

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Singular and Non-Singular Matrices

• A matrix $A$ is singular if $|A|=0$ (no inverse exists).
• A matrix $A$ is non-singular if $|A|\ne 0$ (inverse exists).

Finding $\lambda$ for singularity: Set $|A|=0$ and solve for $\lambda$.

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Adjoint and Inverse of a Matrix

The cofactor matrix of $A$ is the matrix whose $(i,j)$ entry is $A_{ij}$.

The Adjoint (or adjugate) is the transpose of the cofactor matrix: $\text{Adj}(A)=[A_{ij}{]}^T$

The inverse of a non-singular matrix is: $A^{-1}=\frac{1}{|A|}\text{Adj}(A),|A|\ne 0$

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MCQ Practice