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Determinents

Issue No 43, 11 September 2023

By: Anthony O. Ives

Determinents are a set of numbers arranged in rows and columns similar to matrices with the exception that a determinent has to have the same number of rows and columns which is the same as square matrices. The number of rows and columns is the order of the determinent so therefore a 2nd order determinent will have 2 rows and 2 columns, a 3rd order determinent will have 3 rows and 3 columns, etc. Determinents like matrices can be any order but as they are generally used by computer to solve simultaneous equations [1] therefore will be limited as to highest order of determinent due to computing resources. Matices and determinents are used by computers to solve other problems as well but their biggest application is probably solving simultaneous equations. Matrices will discussed in more detail in later articles.

Determinents use a process to determine a single value from its set of numbers with involves multiplying the numbers in the determinent in specific order with a specific sign convention. We will start with 2nd order determinents which are the easiest to work with without a computer and then look at 3rd order determinent which are slightly more complicated and time consuming to evaluate. The process of evaluating a determinent is actually quite tedious for a human being and matrices in generally are like this but an understanding of process allows us to program a computer to do it for any order of determinent or matrix.

The process for evaluating a 2nd order determinent is given below:

\[\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} = a_1b_2 - a_2b_1\]

You may have noticed that the pattern here is you multply the diagonals and add them together with the complication that the product of one of the diagonals must have a negative sign. The other important thing to note is you only use each number in the determine once. You can apply these princples when evaluating a determinent of any order. Simultaneous equations where introduced in a previous article so before explaining the process for evaluating a 3rd order determinent. We will look at how you use 2nd order determinents to solve simultaneous equations with 2 unknowns.

Equation (1): \(a_1x + b_1y + d_1 = 0\)

Equation (2): \(a_2x + b_2y + d_2 = 0\)

\[\Delta_0 = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} \]

\[\Delta_1 = \begin{vmatrix} b_1 & d_1 \\ b_2 & d_2 \end{vmatrix} \]

\[\Delta_2 = \begin{vmatrix} a_1 & d_1 \\ a_2 & d_2 \end{vmatrix} \]

\[\frac{x}{\Delta_1} = -\frac{y}{\Delta_2} = \frac{1}{\Delta_0} \]

You can now use this process to solve the example given in the previous article on simultaneous equations [1].

Equation (1): \(2x + 3y = 4\)

Equation (2): \(x + 6y = 2\)

Equation (1): \(2x + 3y -4 = 0\)

Equation (2): \(x + 6y -2 = 0\)

\[\Delta_0 = \begin{vmatrix} 2 & 3 \\ 1 & 6 \end{vmatrix} \]

\[\Delta_0 = (2 \times 6) - (1 \times 3) = 12 - 3 = 9\]

\[\Delta_1 = \begin{vmatrix} 3 & -4 \\ 6 & -2 \end{vmatrix} \]

\[\Delta_1 = (3 \times -2) - (6 \times -4) = -6 + 24 = 18\]

\[\Delta_2 = \begin{vmatrix} 2 & -4 \\ 1 & -2 \end{vmatrix} \]

\[\Delta_2 = (2 \times -2) - (1 \times -4) = -4 + 4 = 0\]

\[\frac{x}{18} = -\frac{y}{0} = \frac{1}{9} \]

\( x = 18 \times \frac{1}{9} = 2\), \( y = 0 \times \frac{1}{9} = 0\)

As you can see the determinent approach to solving simultaneous equations is more systematic which is ideal for computers but slightly more tedious for a human being to do. Computers are generally very good at doing lots of simple tedious calculations where as a human being is more likely to get bored and loose concentration. So over time clever human beings have devised methods to use a computer's ability to lots of simple tedious calculations to solve complex problems, but in reality computers are quite dumb its how human beings use and program computer is what makes them look clever. The process for evaluating a 3rd order determinent is given below:

\[ \begin{vmatrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{vmatrix} \]

\[ = a_1\begin{vmatrix} b_2 & c_2 \\ b_3 & c_3 \end{vmatrix} - b_1\begin{vmatrix} a_2 & c_2 \\ a_3 & c_3 \end{vmatrix} + c_1\begin{vmatrix} a_2 & b_2 \\ a_3 & b_3 \end{vmatrix} \]

\[ = a_1(b_2c_3-b_3c_2)-b_1(a_2c_3-a_3c_2)+c_1(a_2b_3-a_3b_2) \]

\[\begin{vmatrix} + & - & + \\ - & + & - \\ + & - & + \end{vmatrix} \]

The above method of evaluating a determinent is probably the simplest one however, you could use a number of processes to evaluate a determinent which multiplies each element along a diagonal while using a correct sign convention, of course the diagonal pattern will not always be straight but could follow a zig-zag pattern sometimes. You could also think of it as selecting one element and then creating determinent from the remaining corresponding elements however, the method described above is probably the least confusing and should allow you to spot the pattern which would allow you to evaluate higher order determinents. The process to using 3rd order determinents to solve simultaneous equations with 3 unknowns is similar to that using 2nd order determinents to solve simultaneous equations with 2 unknowns.

Equation (1): \(a_1x + b_1y + c_1z + d_1 = 0\)

Equation (2): \(a_2x + b_2y + c_2z + d_2 = 0\)

Equation (2): \(a_3x + b_3y + c_3z + d_3 = 0\)

\[\Delta_0 = \begin{vmatrix} a_1 & b_1 & c_1\\a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{vmatrix} \]

\[\Delta_1 = \begin{vmatrix} b_1 & c_1 & d_1\\b_2 & c_2 & d_2\\ b_3 & c_3 & d_3 \end{vmatrix} \]

\[\Delta_2 = \begin{vmatrix} a_1 & c_1 & d_1\\a_2 & c_2 & d_2\\a_3 & c_3 & d_3 \end{vmatrix} \]

\[\Delta_3 = \begin{vmatrix} a_1 & b_1 & d_1\\a_2 & b_2 & d_2\\a_3 & b_3 & d_3 \end{vmatrix} \]

\[\frac{x}{\Delta_1} = -\frac{y}{\Delta_2} = \frac{z}{\Delta_3} = -\frac{1}{\Delta_0} \]

You can now use this process to solve the example for simultaneous equations with 3 unknowns given in the earlier article [1].

Equation (1): \(2x + 3y + 5z= 4\)

Equation (2): \(x + 6y + 3z = 2\)

Equation (3): \(4x + 5y + 7z = 6\)

Equation (1): \(2x + 3y + 5z -4 = 0\)

Equation (2): \(x + 6y + 3z -2 = 0\)

Equation (3): \(4x + 5y + 7z -6 = 0\)

\[\Delta_0 = \begin{vmatrix} 2 & 3 & 5 \\ 1 & 6 & 3 \\ 4 & 5 & 7 \end{vmatrix} \]

\[\begin{vmatrix} 2 & | & | \\ | & 6 & 3 \\ | & 5 & 7 \end{vmatrix} = 2\begin{vmatrix} 6 & 3 \\ 5 & 7 \end{vmatrix} \]

\[\begin{vmatrix} | & 3 & | \\ 1 & | & 3 \\ 4 & | & 7 \end{vmatrix} = -3\begin{vmatrix} 1 & 3 \\ 4 & 7 \end{vmatrix} \]

\[\begin{vmatrix} | & | & 5 \\ 1 & 6 & | \\ 4 & 5 & | \end{vmatrix} = 5\begin{vmatrix} 1 & 6 \\ 4 & 5 \end{vmatrix} \]

\[ = +2\begin{vmatrix} 6 & 3 \\ 5 & 7 \end{vmatrix} - 3\begin{vmatrix} 1 & 3 \\ 4 & 7 \end{vmatrix} + 5\begin{vmatrix} 1 & 6 \\ 4 & 5 \end{vmatrix} \]

\[\begin{vmatrix} 6 & 3 \\ 5 & 7 \end{vmatrix} = (6 \times 7) - (5 \times 3) = 42 - 15 = 27\]

\[\begin{vmatrix} 1 & 3 \\ 4 & 7 \end{vmatrix} = (1 \times 7) - (4 \times 3) = 7 - 12 = -5\]

\[\begin{vmatrix} 1 & 6 \\ 4 & 5 \end{vmatrix} = (1 \times 5) - (4 \times 6) = 5 - 24 = -19\]

\[(2 \times 27) - (3 \times -5) + (5 \times -19) = 54 + 15 - 95 = -26\]

\[\Delta_1 = \begin{vmatrix} 3 & 5 & -4 \\ 6 & 3 & -2 \\ 5 & 7 & -6 \end{vmatrix} \]

\[\Delta_2 = \begin{vmatrix} 2 & 5 & -4 \\ 1 & 3 & -2 \\ 4 & 7 & -6 \end{vmatrix} \]

\[\Delta_3 = \begin{vmatrix} 2 & 3 & -4 \\ 1 & 6 & -2 \\ 4 & 5 & -6 \end{vmatrix} \]

As you can see evaluating simulaneous equations with 3 unknowns is long winded and tedious so that is why I will leave it as action point for you to do and of course you can check you answer in the article I did on simultaneous equations. I hope this has demonstrated the process of evaluating determinents and using them to solve simultaneous equations. This process is the absolute way to evaluate a determinent but in a later article I will look at using Gaussian elimination with uses a similar process similar to that used in previous article [1] on simultaneous equations. I have found that the absolute way of solving determinents can be quite computer intensive where as you can still program a computer to solve determinents using Gaussian elimination which for me seemed to be less computer intensive. Determinents are also important for evaluating the inverse of matrix, so in later articles determinents and matrices will be discussed further, you can however, find more information on matrices and determinents in engineering mathematics textbooks one which I have always used is the series written by K. A. Stroud [2] and [3]. Let me know via a comment or email how you get on using 3rd order determinents to solve simultaneous equations.

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References:

[1] http://www.eiteog.com/EiteogBLOG/No40EiteogBlogSimEquations.html

[2] Engineering Mathematics, K. A. Stroud, Fourth Edition, 1995, Macmillan

[3] Further Engineering Mathematics, K. A. Stroud, Fourth Edition, 1996, Palgrave Macmillan

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