Determinants-System of Linear Equations

Relationship Between Determinants and Systems of Equations:

1. Matrix Representation:

Consider a system of linear equations:

${a}_{11}{x}_{1}+{a}_{12}{x}_{2}+\cdots +{a}_{1n}{x}_{n}={b}_{1}$

${a}_{21}{x}_{1}+{a}_{22}{x}_{2}+\cdots +{a}_{2n}{x}_{n}={b}_{2}$

${a}_{n1}{x}_{1}+{a}_{n2}{x}_{2}+\cdots +{a}_{nn}{x}_{n}={b}_{n}$

This system can be represented in matrix form as $AX=B$ where:

• $A$ is the coefficient matrix containing coefficients ${a}_{ij}$.
• $X$ is the column matrix of variables ${x}_{i}$.
• $B$ is the column matrix of constants ${b}_{i}$.

2. Determinants and Solutions:

• Cramer's Rule: For a system of equations represented by $AX=B$, the solution for ${x}_{i}$ can be expressed in terms of determinants of matrices derived from $A$ by replacing the $i$th column with matrix $B$.

• Non-Zero Determinant: If the determinant of matrix $A$($\mathrm{\mid }A\mathrm{\mid }$) is non-zero, the system has a unique solution. Each variable ${x}_{i}$ can be determined uniquely.

• Zero Determinant: If $\mathrm{\mid }A\mathrm{\mid }=0$:

• The system may have infinitely many solutions or no solutions at all.
• The columns of matrix $A$ are linearly dependent, indicating redundancy or inconsistency in the equations.

Example:

Let's consider a system of linear equations:

$2x+3y=8$

$4x-2y=2$

Representing this system in matrix form:

$A=\left[\begin{array}{cc}2& 3\\ 4& -2\end{array}\right]$

$X=\left[\begin{array}{c}x\\ y\end{array}\right]$

$B=\left[\begin{array}{c}8\\ 2\end{array}\right]$

• Calculate the determinant of matrix $A$ ($\mathrm{\mid }A\mathrm{\mid }$).
• Determine if the system has a unique solution based on the value of $\mathrm{\mid }A\mathrm{\mid }$.
• Apply Cramer's Rule to find the solution if $\mathrm{\mid }A\mathrm{\mid }$ is non-zero.

Cramer's Rule:

Given a system of linear equations:

${a}_{11}{x}_{1}+{a}_{12}{x}_{2}+\cdots +{a}_{1n}{x}_{n}={b}_{1}$

${a}_{21}{x}_{1}+{a}_{22}{x}_{2}+\cdots +{a}_{2n}{x}_{n}={b}_{2}$

${a}_{n1}{x}_{1}+{a}_{n2}{x}_{2}+\cdots +{a}_{nn}{x}_{n}={b}_{n}$

Represented in matrix form: $AX=B$

Where:

• $A$ is the coefficient matrix containing coefficients ${a}_{ij}$.
• $X$ is the column matrix of variables ${x}_{i}$.
• $B$ is the column matrix of constants ${b}_{i}$.

Using Cramer's Rule:

1. Calculating Determinants:

• Compute the determinant of matrix $A$ denoted as $\mathrm{\mid }A\mathrm{\mid }$.
• For each variable ${x}_{i}$, replace the $i$th column of matrix $A$ with matrix $B$ to create a new matrix, say ${A}_{i}$.
• Calculate the determinant of matrix ${A}_{i}$ denoted as $\mathrm{\mid }{A}_{i}\mathrm{\mid }$.
2. Solving for Variables:

• The solution for variable ${x}_{i}$ is given by: ${x}_{i}=\frac{\mathrm{\mid }{A}_{i}\mathrm{\mid }}{\mathrm{\mid }A\mathrm{\mid }}$
3. Unique Solution:

• If $\mathrm{\mid }A\mathrm{\mid }\mathrm{\ne }0$, Cramer's Rule provides a unique solution for each variable ${x}_{i}$.
• If $\mathrm{\mid }A\mathrm{\mid }=0$ or $\mathrm{\mid }{A}_{i}\mathrm{\mid }=0$ for some $i$, Cramer's Rule cannot be applied or the system may have no unique solution.

Example:

Let's solve the system of linear equations using Cramer's Rule:

$3x+2y=11$

$2x-y=3$

Representing this system in matrix form:

$A=\left[\begin{array}{cc}3& 2\\ 2& -1\end{array}\right]$ $X=\left[\begin{array}{c}x\\ y\end{array}\right]$ $B=\left[\begin{array}{c}11\\ 3\end{array}\right]$

1. Calculate the determinant of matrix $A$ ($\mathrm{\mid }A\mathrm{\mid }$).
2. Calculate determinants $\mathrm{\mid }{A}_{1}\mathrm{\mid }$ and $\mathrm{\mid }{A}_{2}\mathrm{\mid }$.
3. Use Cramer's Rule to find $x$ and $y$.