find the value of X ,Y and Z with using crammer's rule.
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Answer:
Cramer’s Rule for a 3×3 System (with Three Variables)
In our previous lesson, we studied how to use Cramer’s Rule with two variables. Our goal here is to expand the application of Cramer’s Rule to three variables usually in terms of x, y, and z. I will go over five (5) worked examples to help you get familiar with this concept.
To do well on this topic, you need to have an idea how to find the determinant of a 3×3 matrix. So, this is what we are going to do first. Ready?
worked examples to help you get familiar with this concept.
To do well on this topic, you need to have an idea how to find the determinant of a 3×3 matrix. So, this is what we are going to do first. Ready?
Formula to Find the Determinant of a 3×3 Matrix
Given a 3×3 matrix
Matrix A is a 3 by 3 square matrix with elemenys a, b and c on the first row, elements d, e and f on the second row, and elements g, h and i on the third row. We can write Matrix in compact for as A = [a,b,c;d,e,f;g,h,i].
Its determinant can be calculated using the following formula.
The determinant of matrix A = [a,b,c;d,e,f;g,h,i] is calculated as as follows: |A| = a times the determinant of matrix [e,f;h,i] minus b times the determinant of matrix [d,f;g,i] plus c times the determinant of matrix [d,e;g,h]. In compact form, the determinat of matrix A is |A| = a*|e,g;h,i| - b*|d,f;g,i| + c*|d,e;g,h|.
Let’s do a quick example on this.
Find the determinant of matrix A
Matrix A is a 3x3 square matrix with entries 6, 2 and -4 on its first row, entries 5, 6 and -2 on its second row, and entries 5,2 and -3 on its third row. Therefore we can write matrix A as A = [6,2,-4;5,6,-2;5,2;-3].
Solution: Make sure that you follow the formula on how to find the determinant of a 3×3 matrix carefully, as shown above. More so, don’t rush when you perform the required arithmetic operations in every step. This is where common errors usually occur, but it can be prevented. When you do it right, your solution should be similar to the one below.
To find the determinant of the square (3 by 3) matrix A = [6,2,-4;5,6,-2;5,2,-3], we have the following steps: |A|=|6,2,-4;5,6,-2;5,2,-3|=6*|6,-2;2,-3| - (2)*|5,-2;5,-3|+(-4)*|5,6;5,2| = 6(-14) - 2(-5) - 4(-20) = -84 +10 + 80 = 6. Therefore, the determinant of matrix A is equal to 6.
Now, it’s time to go over the procedure on how to use Cramer’s Rule in a linear system involving three variables.
Cramer’s Rules for Systems of Linear Equations with Three Variables
