Algebra 2 - Determinants & Cramer's Rule

Introduction

• A determinant is an array of numbers consisting of row elements and column elements that are typically used to solve systems of linear equations. 
• In the symbol $\left|A_n\right|=\left|\begin{array}{cc}{a}_{11}& a_{12}...a_{1n}\\ a_{21}& a_{22}...a_{2n}\\ .& . .\\ .& . .\\ a_{n1}& a_{n2...}{a}_{nn}\end{array}\right|$, the determinant of order $\mathit{n}$ is defined.
• This algebraic symbol is employed to express the compact form for the sum of all possible products wherein each has $\mathit{n}$ factors. 
• Each element is contained with two subscripts, the first number indicates the row order in which the element is arranged, and the second number denotes the column order in which the element is arranged.

Order and Evaluating Determinants

• The second order of determinant (also 2 x 2 determinant) is in the form of $\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|$.
• To illustrate, this is defined as:

$\mathbf{\left|}{\mathbf{A}}_{\mathbf{2}}\mathbf{\right|}\mathbf{=}\mathbf{\left|}\begin{array}{cc}{\mathbf{a}}_{\mathbf{11}}& {\mathbf{a}}_{\mathbf{12}}\\ {\mathbf{a}}_{\mathbf{21}}& {\mathbf{a}}_{\mathbf{22}}\end{array}\mathbf{\right|}\mathbf{=}{\mathit{a}}_{\mathbf{11}}{\mathit{a}}_{\mathbf{22}}\mathbf{-}{\mathit{a}}_{\mathbf{12}}{\mathit{a}}_{\mathbf{21}}$

• For instance, if we have $\mathrm{\left|}\begin{array}{cc}\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{8}& \mathrm{-}\mathrm{7}\\ \mathrm{-}\mathrm{6}& \mathrm{-}\mathrm{4}\end{array}\mathrm{\right|}$, this is evaluated as:

         $=\left(8\right)\left(-4\right)-\left(-7\right)\left(-6\right)$

         $=-32-42$

         $=-74$

• The third order of determinant is described as:

$A_3=\left|\begin{array}{ccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& {\mathrm{a}}_{13}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}& {\mathrm{a}}_{23}\\ {\mathrm{a}}_{31}& {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right|$

• Rewrite this first and second column in order to evaluate the third-order determinant:

$A_3=\left|\begin{array}{ccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& {\mathrm{a}}_{13}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}& {\mathrm{a}}_{23}\\ {\mathrm{a}}_{31}& {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right|\begin{array}{c}{\mathrm{a}}_{11}\\ {\mathrm{a}}_{21}\\ {\mathrm{a}}_{31}\end{array}\begin{array}{c}{\mathrm{a}}_{12}\\ {\mathrm{a}}_{22}\\ {\mathrm{a}}_{32}\end{array}$

$A_3=a_{11}{a}_{22}{a}_{33}+a_{12}{a}_{23}{a}_{31}+a_{13}{a}_{21}{a}_{32}-a_{31}{a}_{22}{a}_{13}-a_{32}{a}_{23}{a}_{11}-a_{33}{a}_{21}{a}_{12}$

Basic Properties of Determinants

• When corresponding rows and columns are interchanged, it does not affect the value of the determinant. 

Examples:

$\left|\begin{array}{ccc}4& 9& 4\\ 5& 3& 5\\ 1& 8& 0\end{array}\right|=\left|\begin{array}{ccc}4& 5& 1\\ 9& 3& 8\\ 4& 5& 0\end{array}\right| and \left|\begin{array}{ccc}6& 7& 10\\ 9& 2& 1\\ 3& 4& 5\end{array}\right|=\left|\begin{array}{ccc}6& 9& 3\\ 7& 2& 4\\ 10& 1& 5\end{array}\right|$

• Any two rows or columns interchanged alters the sign of the determinant.

Examples:

$\left|\begin{array}{ccc}6& 7& 4\\ 2& 5& 3\\ 3& 0& -2\end{array}\right|=-\left|\begin{array}{ccc}2& 5& 3\\ 6& 7& 4\\ 3& 0& -2\end{array}\right| and \left|\begin{array}{ccc}5& 6& 8\\ 2& 3& 4\\ 1& 5& -2\end{array}\right|=\left|\begin{array}{ccc}5& 6& 8\\ 1& 5& -2\\ 2& 3& 4\end{array}\right|$

• Suppose that all elements in a row or column are zero; then the value of this determinant is 0.

Examples:

$\left|\begin{array}{ccc} 3& 0& 7\\ -1& 0& 8\\ 5& 0& 9\end{array}\right|=0 and \left|\begin{array}{ccc}3& 4& 7\\ 0& 0& 0\\ 7& 6& 5\end{array}\right|=0$

• If there are any two rows or columns that are identical, the value of this determinant is equal to 0.

Examples:

$\left|\begin{array}{ccc}5& 4& 4\\ 3& -3& -3\\ 9& 2& 2\end{array}\right|=0 and \left|\begin{array}{ccc}3& 4& 2\\ 6& 7& 4\\ 6& 7& 4\end{array}\right|=0$

• If each element in a row or column is multiplied by any number $\mathit{n}$, the value of the entire determinant is also multiplied by the number $\mathit{n}\mathbf{.}$

Example:

$\left(3\right)\left|\begin{array}{ccc}7& 2& 6\\ 4& 7& 2\\ 5& 3& 8\end{array}\right|=\left|\begin{array}{ccc}21& 6& 18\\ 12& 21& 6\\ 15& 9& 24\end{array}\right|$

• If each element in a row or column of a determinant is expressed as a sum of two or more numbers, the determinant can be written as a sum of two or more determinants.

Example:

$\left|\begin{array}{ccc} 8& 0& 3\\ 4& 9& 8\\ -1& 7& 4\end{array}\right|=\left|\begin{array}{ccc} 5+3& 0& 3\\ 2+2& 9& 8\\ -4+3& 7& 4\end{array}\right|=\left|\begin{array}{ccc}5& 0& 3\\ 2& 9& 8\\ -4& 7& 4\end{array}\right|+\left|\begin{array}{ccc}3& 0& 3\\ 2& 9& 8\\ 3& 7& 4\end{array}\right|\begin{array}{}\\ \\ \\ \end{array}$

• Suppose that each element of a row or column is being added $\mathit{n}$ times the corresponding elements of another row or column; the value of the determinant will not change.

Example:

$\left|\begin{array}{ccc}5& 9& 3\\ 6& 8& 7\\ 7& 4& 2\end{array}\right|=\left|\begin{array}{ccc}5+4\left(9\right)& 9& 3\\ 6+4\left(8\right)& 8& 7\\ 7+4\left(4\right)& 4& 2\end{array}\right|=\left|\begin{array}{ccc}5& 9& 3\\ 6& 8& 7\\ 7& 4& 2\end{array}\right|$

Determinant of a Matrix

• To evaluate the determinant of a matrix, the matrix has to be a square matrix. 
• The determinant of a matrix is useful in calculus applications, determining the inverse of a matrix, and solving systems of linear equations.
• To illustrate, we have:

$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right], then detA=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc$

• For a 3 x 3 matrix, we have the following:

$B=\left[\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right]=\left(a_1\right)det\left[\begin{array}{cc}{b}_2& c_2\\ b_3& c_3\end{array}\right]-\left(b_1\right)det\left[\begin{array}{cc}{a}_2& c_2\\ a_3& c_3\end{array}\right]+\left(c_1\right)det\left[\begin{array}{cc}{a}_2& b_2\\ a_3& b_3\end{array}\right]$

Cramer's Rule

• Cramer's Rule is a procedure that is used to solve for the solutions of a system of linear equations in $\mathit{n}$ unknowns.
• To illustrate the process of solving a system of linear equations in two unknowns, we have:
• Given the system of linear equations $\{\begin{array}{l}{a}_{1}x+b_{1}y=c_1\\ a_{2}x+b_{2}y=c_2\end{array}$, then the solution is determined using the following:

      $x=\frac{\left|\begin{array}{cc}{\mathrm{c}}_1& {\mathrm{b}}_1\\ {\mathrm{c}}_2& {\mathrm{b}}_2\end{array}\right|}{\left|\begin{array}{cc}{\mathrm{a}}_1& {\mathrm{b}}_1\\ {\mathrm{a}}_2& {\mathrm{b}}_2\end{array}\right|} and y=\frac{\left|\begin{array}{cc}{\mathrm{a}}_1& {\mathrm{c}}_1\\ {\mathrm{a}}_2& {\mathrm{c}}_2\end{array}\right|}{\left|\begin{array}{cc}{\mathrm{a}}_1& {\mathrm{b}}_1\\ {\mathrm{a}}_2& {\mathrm{b}}_2\end{array}\right|}$ where $\left|\begin{array}{cc}{a}_1& b_1\\ a_2& b_2\end{array}\right|\ne 0$

• This can also be simplified using the following formulas:

      $x=\frac{{\mathrm{b}}_2{\mathrm{c}}_1-{\mathrm{b}}_1{\mathrm{c}}_2}{{\mathrm{a}}_1{\mathrm{b}}_2-{\mathrm{a}}_2{\mathrm{b}}_1} and y=\frac{{\mathrm{a}}_1{\mathrm{c}}_2-{\mathrm{a}}_2{\mathrm{c}}_1}{{\mathrm{a}}_1{\mathrm{b}}_2-{\mathrm{a}}_2{\mathrm{b}}_1}$ where $a_1{b}_2-a_2{b}_1\ne 0$

• To illustrate the process of solving a system of linear equations in three unknowns, we have:

        1. Given the linear system $\{\begin{array}{l}{a}_{1}x+b_{1}y+c_{1}z=d_1\\ a_{2}x+b_{2}y+c_{2}z=d_2\\ a_{3}x+b_{3}y+c_{3}z=d_3\end{array}$, then this is expressed as a determinant $\mathit{D}\mathbf{,}\mathbf{ }{\mathit{D}}_{\mathbf{x}}\mathbf{,}\mathbf{ }\mathit{a}\mathit{n}\mathit{d}\mathbf{ }{\mathit{D}}_{\mathbf{y}}$.

        2. We compute the following determinants in order to find the solution to the given linear system with three unknowns.

    $D=\left|\begin{array}{ccc}{\mathrm{a}}_1& {\mathrm{b}}_1& {\mathrm{c}}_1\\ {\mathrm{a}}_2& {\mathrm{b}}_2& {\mathrm{c}}_2\\ {\mathrm{a}}_3& {\mathrm{b}}_2& {\mathrm{c}}_3\end{array}\right|\begin{array}{c}{\mathrm{a}}_1\\ {\mathrm{a}}_2\\ {\mathrm{a}}_3\end{array}\begin{array}{c}{\mathrm{b}}_1\\ {\mathrm{b}}_2\\ {\mathrm{b}}_3\end{array}, D_{\mathrm{x}}=\left|\begin{array}{ccc}{\mathrm{d}}_1& {\mathrm{b}}_1& {\mathrm{c}}_1\\ {\mathrm{d}}_2& {\mathrm{b}}_2& {\mathrm{c}}_2\\ {\mathrm{d}}_3& {\mathrm{b}}_3& {\mathrm{c}}_3\end{array}\right|\begin{array}{c}{\mathrm{d}}_1\\ {\mathrm{d}}_2\\ {\mathrm{d}}_3\end{array}\begin{array}{c}{\mathrm{b}}_1\\ {\mathrm{b}}_2\\ {\mathrm{b}}_3\end{array}, D_{\mathrm{y}}=\left|\begin{array}{ccc}{\mathrm{a}}_1& {\mathrm{d}}_1& {\mathrm{c}}_1\\ {\mathrm{a}}_2& {\mathrm{d}}_2& {\mathrm{c}}_2\\ {\mathrm{a}}_3& {\mathrm{d}}_3& {\mathrm{c}}_3\end{array}\right|\begin{array}{c}{\mathrm{a}}_1\\ {\mathrm{a}}_2\\ {\mathrm{a}}_3\end{array}\begin{array}{c}{\mathrm{d}}_1\\ {\mathrm{d}}_2\\ {\mathrm{d}}_3\end{array}, D_{\mathrm{z}}=\left|\begin{array}{ccc}{\mathrm{a}}_1& {\mathrm{b}}_1& {\mathrm{d}}_1\\ {\mathrm{a}}_2& {\mathrm{b}}_2& {\mathrm{d}}_2\\ {\mathrm{a}}_3& {\mathrm{b}}_3& {\mathrm{d}}_3\end{array}\right|\begin{array}{c}{\mathrm{a}}_1\\ {\mathrm{a}}_2\\ {\mathrm{a}}_3\end{array}\begin{array}{c}{\mathrm{b}}_1\\ {\mathrm{b}}_2\\ {\mathrm{b}}_3\end{array}$

       3. After evaluating the determinants above, the following equations will be used to find the solution: $x=\frac{{\mathrm{D}}_{\mathrm{x}}}{\mathrm{D}}, y=\frac{{\mathrm{D}}_{\mathrm{y}}}{\mathrm{D}}, and z=\frac{{\mathrm{D}}_{\mathrm{z}}}{\mathrm{D}}$

Berzen's Layer Reduction Method

• Another method of solving systems of linear equations in three unknowns is Berzen's Layer Reduction Method.
• Given a linear system $\{\begin{array}{l}{a}_{1}x+b_{1}y+c_{1}z=d_1\\ a_{2}x+b_{2}y+c_{2}z=d_2\\ a_{3}x+b_{3}y+c_{3}z=d_3\end{array}$, then we solve the following determinants: $A, B, C, D, E, and F$.

       $A=\left|\begin{array}{cc}{\mathrm{a}}_1& {\mathrm{b}}_1\\ {\mathrm{a}}_2& {\mathrm{b}}_2\end{array}\right|, B=\left|\begin{array}{cc}{\mathrm{a}}_1& {\mathrm{c}}_1\\ {\mathrm{a}}_2& {\mathrm{c}}_2\end{array}\right|, C=\left|\begin{array}{cc}{\mathrm{a}}_1& {\mathrm{d}}_1\\ {\mathrm{a}}_2& {\mathrm{d}}_2\end{array}\right|, D=\left|\begin{array}{cc}{\mathrm{a}}_2& {\mathrm{b}}_2\\ {\mathrm{a}}_3& {\mathrm{b}}_3\end{array}\right|, E=\left|\begin{array}{cc}{\mathrm{a}}_2& {\mathrm{c}}_2\\ {\mathrm{a}}_3& {\mathrm{c}}_3\end{array}\right|, F=\left|\begin{array}{cc}{\mathrm{a}}_2& {\mathrm{d}}_2\\ {\mathrm{a}}_3& {\mathrm{d}}_3\end{array}\right|$

• After evaluating the above determinants, we need to find the value of $\mathit{z}$ using the formula below. Simply direct substitute the obtained values and simplify.

      $z=\frac{\mathrm{AF}-\mathrm{DC}}{\mathrm{AE}-\mathrm{BD}}$ where $AE-BD\ne 0$

• After getting the value of $\mathit{z}$, the next step is to substitute the computed value of $z$ in any of the two equations and a linear system of two variables will be framed. Solve for the value of $\mathit{x}$ and $\mathit{y}$ using any appropriate algebraic procedure.

Cheat Sheet

• In case of $\mathrm{\{}\begin{array}{l}\mathrm{x}\mathrm{+}\mathrm{y}\mathrm{=}\mathrm{1}\\ \mathrm{2}\mathrm{x}\mathrm{+}\mathrm{3}\mathrm{y}\mathrm{=}\mathrm{6}\end{array}$, then $D=\left|\begin{array}{cc}1& 1\\ 2& 3\end{array}\right|, D_{\mathrm{x}}=\left|\begin{array}{cc}1& 1\\ 6& 3\end{array}\right|, D_{\mathrm{y}}=\left|\begin{array}{cc}1& 1\\ 2& 6\end{array}\right|$
• In a system of the linear equation of two variables, we have the following equations: $x=\frac{{\mathrm{D}}_{\mathrm{x}}}{\mathrm{D}} and y=\frac{{\mathrm{D}}_{\mathrm{y}}}{\mathrm{D}}$.
• In case of $A=\left|\begin{array}{ccc} 7& 3& 6\\ 4& 9& -2\\ -5& 7& 8\end{array}\right|=\left|\begin{array}{ccc} 7& 3& 6\\ 4& 9& -2\\ -5& 7& 8\end{array}\right|\begin{array}{c} 7\\ 4\\ -5\end{array}\begin{array}{c} 3\\ 9\\ 7\end{array}$, we compute the determinant as shown below:

      $=\left(7\right)\left(9\right)\left(8\right)+\left(3\right)\left(-2\right)\left(-5\right)+\left(6\right)\left(4\right)\left(7\right)$$-\left(-5\right)\left(9\right)\left(6\right)-\left(7\right)\left(-2\right)\left(7\right)-\left(8\right)\left(4\right)\left(3\right)$

     $=504+30+168-\left(-270\right)-\left(-98\right)-96$

     $=974$

• In solving a linear system of three unknowns, we have $x=\frac{{\mathrm{D}}_{\mathrm{x}}}{\mathrm{D}}, y=\frac{{\mathrm{D}}_{\mathrm{y}}}{\mathrm{D}}, and z=\frac{{\mathrm{D}}_{\mathrm{z}}}{\mathrm{D}}$
• In case of $D=\left[\begin{array}{ccc}1& 0& 2\\ 1& 1& -2\\ 3& -1& 1\end{array}\right],$ we can evaluate the determinant of this matrix by following the procedure below:

     $=\left(1\right)det\left[\begin{array}{cc}1& -2\\ -1& 1\end{array}\right]-\left(0\right)det\left[\begin{array}{cc}1& -2\\ 3& 1\end{array}\right]+\left(2\right)det\left[\begin{array}{cc}1& 1\\ 3& -1\end{array}\right]$

Blunder Areas

• Do not be confused with the symbol used in determinant of a matrix and absolute value. The determinant of a matrix is not an absolute value symbol. 
• When using Cramer's Rule in solving linear systems, the process may fail if the determinant of the coefficient array is zero.
• Cramer's Rule only applies to a square matrix wherein that system has a unique solution and has a non-zero determinant. 
• Berzen's Layer Reduction Method only works for a linear system of three variables.