How to Solve Systems of Linear Equations Using Determinants

A system of linear equations using determinants refers to the algebraic process of representing and solving linear systems by expressing them in matrix form, then applying determinant-based methods such as Cramer's Rule to find solutions for variables.

Matrix Representation and Determinant Structure of Linear Systems

A system of $n$ linear equations in $n$ variables can be written as:

$ \begin{aligned} a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n &= b_1 \\ a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n &= b_2 \\ \vdots \hspace{3cm} \vdots \\ a_{n1}x_1 + a_{n2}x_2 + \cdots + a_{nn}x_n &= b_n \end{aligned} $

This system is compactly expressed as $A\vec{x} = \vec{b}$, where $A$ is the coefficient matrix, $\vec{x}$ is the column vector of variables, and $\vec{b}$ is the constants column vector.

The determinant of the coefficient matrix governs the nature of solutions, as determined by properties outlined in advanced matrix theory (Matrices And Determinants).

Cramer's Rule for Solving Systems Using Determinants

For a system $A\vec{x} = \vec{b}$ where $A$ is a non-singular square matrix of order $n$ (i.e., $|A| \neq 0$), Cramer's Rule yields the unique solution:

$ x_i = \dfrac{|\Delta_i|}{|A|},\quad \text{for } i = 1,2,\ldots,n $

Here, $|\Delta_i|$ denotes the determinant obtained by replacing the $i$-th column of $A$ with the column vector $\vec{b}$. Thus, calculations involve $n+1$ determinants of order $n$.

Classification of Solution Sets via Determinants

The analysis of $|A|$ provides the trichotomy for a square system:

• Unique solution if $|A| \neq 0$
• No solution or infinitely many solutions if $|A| = 0$
• Infinitely many solutions if $|A| = 0$ and all $|\Delta_i| = 0$

If $|A| = 0$ but at least one $|\Delta_i| \neq 0$, the system is inconsistent and has no solution. This manifests as parallel lines or planes not meeting at a common point.

Component Calculation for Two and Three Variable Systems

Consider the two-variable system:

$ \begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \\ \end{cases} $

Define $ D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix},\ D_x = \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix},\ D_y = \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}. $ Then $ x = \dfrac{D_x}{D},\ y = \dfrac{D_y}{D} $ when $D \neq 0$.

For a three-variable system as in:

$ \begin{cases} a_1x + b_1y + c_1z = d_1 \\ a_2x + b_2y + c_2z = d_2 \\ a_3x + b_3y + c_3z = d_3 \\ \end{cases} $

Let $D$, $D_x$, $D_y$, $D_z$ be similarly constructed $3\times 3$ determinants. Then $x = \frac{D_x}{D}$, $y = \frac{D_y}{D}$, $z = \frac{D_z}{D}$ if $D \neq 0$.

Illustrative Solutions of Linear Systems Using Determinant Methods

Example: Solve by determinants: $2x + 3y = 8$, $x - y = 1$

Compute $D = \begin{vmatrix} 2 & 3 \\ 1 & -1 \end{vmatrix} = (2)(-1)-(1)(3) = -2-3 = -5$.

Compute $D_x = \begin{vmatrix} 8 & 3 \\ 1 & -1 \end{vmatrix} = (8)(-1)-(1)(3) = -8-3 = -11$.

Compute $D_y = \begin{vmatrix} 2 & 8 \\ 1 & 1 \end{vmatrix} = (2)(1)-(1)(8) = 2-8 = -6$.

Solution: $x = \dfrac{-11}{-5} = {11\over 5}$ , $y = \dfrac{-6}{-5} = {6\over 5}$.

Example: Test for consistency:
$x + 2y + 3z = 6$
$2x + 3y + 2z = 5$
$3x + y + z = 4$

Compute $D = \begin{vmatrix} 1 & 2 & 3\\ 2 & 3 & 2\\ 3 & 1 & 1 \end{vmatrix} $

Expand determinant column-wise:

$1\cdot[(3)(1)-(2)(1)] - 2\cdot[(2)(1)-(3)(3)] + 3\cdot[(2)(1)-(3)(3)]$
$= 1(3-2) - 2(2-9) + 3(2-9)$
$= 1(1) - 2(-7) + 3(-7)$
$= 1 +14 -21 = -6$

Since $D \neq 0$, the system is consistent and has a unique solution.

Algebraic Consequences of Determinant Zero in Homogeneous Linear Systems

A homogeneous system ($A\vec{x} = \vec{0}$) always has the trivial solution. If $|A| = 0$, the system possesses infinitely many non-trivial solutions.

This property is crucial in parametric and vector space interpretations of solution sets (System Of Linear Equations Using Determinants).

Rank Conditions for Consistency Using the Matrix Method

For any $m \times n$ system $A\vec{x} = \vec{b}$, the system is consistent if and only if the rank of $A$ equals the rank of the augmented matrix $[A|\vec{b}]$.

If rank$(A) = n$, and $n$ is number of variables, the unique solution exists. For rank$(A) < n$ and consistency, solutions are infinitely many. For further reference, see system of linear equations in matrix form.

Common Analytical Errors in Determinant Approaches to Linear Systems

A frequent source of error is neglecting to check the non-vanishing of the determinant before applying Cramer’s Rule.

Singular matrices (with zero determinant) render Cramer's approach invalid, and one must revert to rank analysis or back-substitution methods (Properties Of Determinants).

Variations of Determinant Methods in Linear System Problem Types

• Non-homogeneous systems with unique solution
• Homogeneous systems with parametric solutions
• Consistent systems with infinitely many solutions
• Systems inconsistent by determinant or rank
• Reduction to row echelon form and determinant calculation
• Direct computation of variable via determinant formulas

JEE Main questions may require any of the above variations, demanding precise determinant and matrix operations in all steps (Solution Of Triangles).