2.4 Determinants and Special Matrices

Properties of Determinants

Determinants have several key properties that allow simplification without full expansion.

1. Identical Rows or Columns

If any two rows (or columns) of a matrix are identical, its determinant is zero.

Reason: Swapping the two identical rows changes the sign of the determinant but leaves it unchanged, so $|A|=-|A|$, giving $|A|=0$.

2. Scalar Multiple of a Row/Column

If one row (or column) is a scalar multiple of another, the determinant is zero (the rows/columns are linearly dependent).

Example: $\left|\begin{array}{ccc}9& 27& 36\\ 18& 54& 24\\ 27& 81& 28\end{array}\right|=0$ because $C_2=3C_1$.

3. Scalar Factor

If every element of one row (or column) is multiplied by scalar $k$, the determinant is multiplied by $k$: $|k{R}_i|=k|A|$ Conversely, a common factor from any single row or column can be taken outside the determinant.

4. Linearity (Splitting a Row/Column)

If the elements of a row or column are sums, e.g., $c_i=p_i+q_i$, the determinant splits: $\left|\cdots (p_i+q_i)\cdots \right|=\left|\cdots p_i\cdots \right|+\left|\cdots q_i\cdots \right|$

Example: $\left|\begin{array}{ccc}x& x^2& 1+a{x}^3\\ y& y^2& 1+a{y}^3\\ z& z^2& 1+a{z}^3\end{array}\right|=\left|\begin{array}{ccc}x& x^2& 1\\ y& y^2& 1\\ z& z^2& 1\end{array}\right|+a\left|\begin{array}{ccc}x& x^2& x^3\\ y& y^2& y^3\\ z& z^2& z^3\end{array}\right|$

5. Row/Column Operations for Factoring

To factor out $(a+b+c)$ from a determinant, apply $R_1\to R_1+R_2+R_3$. This makes every element of $R_1$ equal to $(a+b+c)$, which is then taken out as a scalar.

Example: To prove $\left|\begin{array}{ccc}x& y& x+y\\ y& x+y& x\\ x+y& x& y\end{array}\right|=-2(x^3+y^3)$ Apply $R_1\to R_1+R_2+R_3$: the first row becomes $[2(x+y),2(x+y),2(x+y)]$, so $2(x+y)$ factors out.

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Special Matrices and Their Determinants

Skew-Symmetric Matrix

A matrix $A$ is skew-symmetric if $A^T=-A$, i.e., $a_{ij}=-a_{ji}$ and all diagonal elements are zero.

Example: $A=\left(\begin{array}{ccc}0& -a& -b\\ a& 0& -c\\ b& c& 0\end{array}\right)$

Key Property: The determinant of every skew-symmetric matrix of odd order is zero.

Proof: Since $A^T=-A$: $|A^T|=|-A|=(-1{)}^n|A|$ For odd $n$: $(-1{)}^n=-1$, so $|A|=-|A|$, giving $|A|=0$.

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Consistent and Inconsistent Systems

A system of linear equations $AX=B$ is:

• Consistent — has at least one solution (unique or infinitely many).
  • Unique solution: $\text{rank}(A)=\text{rank}([A|B])=n$ (number of unknowns)
  • Infinitely many: $\text{rank}(A)=\text{rank}([A|B])<n$
• Inconsistent — has no solution: $\text{rank}(A)\ne \text{rank}([A|B])$

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Solving Non-Homogeneous Systems ($AX=B$)

Matrix Inversion Method

If $|A|\ne 0$, the unique solution is: $X=A^{-1}B,\text{where }{A}^{-1}=\frac{1}{|A|}\text{adj}(A)$

Cramer's Rule

For $AX=B$ with $|A|\ne 0$, the solution is: $x=\frac{|A_x|}{|A|},y=\frac{|A_y|}{|A|},z=\frac{|A_z|}{|A|}$ where $A_x$, $A_y$, $A_z$ are obtained by replacing the respective column of $A$ with $B$.

Example: Solve $x+y+z=6,2x-y+z=3,x+2y-z=2$

Compute $|A|$, then $|A_x|$, $|A_y|$, $|A_z|$ by substituting column $B$ in place of each variable's column.

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Solving Homogeneous Systems ($AX=0$)

A homogeneous system always has the trivial solution $X=0$.

Non-trivial solutions exist if and only if $|A|=0$.

Gaussian Elimination Method

1. Write the augmented matrix $[A|0]$.
2. Apply row operations to reach row echelon form.
3. If a free variable appears (rank $<n$), express other variables in terms of it.
4. Write the general solution.

Example: For $x+y-z=0,2x-y+z=0,x-2y+2z=0$ Form $[A|0]$ and row-reduce. If $|A|=0$, a non-trivial solution exists.