Cramer's Rule Explained

The video begins with an introduction to Cramer's Rule, a method for solving simultaneous equations. The speaker explains that this technique can be applied to matrices of various sizes, including 2x2, 3x3, and even 4x4. The importance of understanding how to calculate determinants is emphasized, as this is a fundamental step in applying Cramer's Rule. The speaker provides a link for viewers to learn about determinants if they are unfamiliar with the concept. The video then proceeds with examples to demonstrate how to find the values of variables given by simultaneous equations.

In this segment, the speaker focuses on solving 2x2 simultaneous equations using Cramer's Rule. The process starts with forming a 2x2 matrix using the coefficients from the equations. The determinant of this matrix (denoted as D) is calculated, which is crucial for the subsequent steps. The speaker then explains how to create two more matrices to find the values of the variables x and y. These new matrices are formed by replacing the coefficients of the variables with the constants from the equations. The determinants of these matrices (DX and DY) are calculated, and the values of the variables are found by dividing DX and DY by D. The speaker provides a worked-out example, demonstrating the steps and calculations involved in solving the equations.

The speaker now extends the application of Cramer's Rule to 3x3 matrices, which involves more complex calculations. The process is similar to the 2x2 case but involves additional steps and a larger matrix. The speaker explains how to form the 3x3 matrix and calculate its determinant (D). The video then details the creation of three new matrices (DX, DY, and DZ) by replacing the coefficients of the variables X, Y, and Z with the constants from the equations. The determinants of these matrices are calculated, and the values of the variables are found by dividing each determinant by D. The speaker provides a detailed example, walking through the calculations and explaining the concept of finding determinants for 3x3 matrices.

This part of the video continues the discussion on solving 3x3 simultaneous equations using Cramer's Rule. The speaker provides a comprehensive walkthrough of the process, including the formation of the 3x3 matrix and the calculation of its determinant (D). The video then moves on to the creation of the three determinants (DX, DY, and DZ) by substituting the coefficients of the variables X, Y, and Z with the constants from the equations. The speaker demonstrates how to calculate these determinants and use them to find the values of the variables x, y, and z. The detailed calculations and step-by-step explanation help viewers understand the process of applying Cramer's Rule to 3x3 matrices.

The speaker concludes the tutorial by reviewing the process of using Cramer's Rule for 3x3 matrices and encourages viewers to practice the method. The speaker provides a set of determinants (D, DX, DY, and DZ) and asks viewers to find the values of the variables X, Y, and Z using the previously explained steps. This interactive approach allows viewers to apply their understanding of the method and check their work against the speaker's forthcoming solutions. The speaker emphasizes the importance of practicing to avoid mistakes and gain proficiency in using Cramer's Rule for solving simultaneous equations.

In the final segment, the speaker presents the correct solutions for the provided determinants and compares them with the solutions that viewers might have found. The speaker's solutions are X=7, Y=-3, and Z=-4, which are derived from the calculated determinants. The speaker invites viewers to ask questions in the comments section if they have any, and concludes the video with a thank you message. This closing section serves as a recap and an opportunity for viewers to clarify any doubts they may have about the application of Cramer's Rule.