We eliminate one variable using row operations and solve for the other. Say that we wish to solve for
If equation (2) is multiplied by the opposite of the coefficient of
in equation (1), equation (1) is multiplied by the coefficient of
in equation (2), and we add the two equations, the variable
will be eliminated.
Now, solve for
Similarly, to solve for
we will eliminate
Solving for
gives
Notice that the denominator for both
and
is the determinant of the coefficient matrix.
We can use these formulas to solve for
and
but Cramer’s Rule also introduces new notation:
determinant of the coefficient matrix
determinant of the numerator in the solution of
determinant of the numerator in the solution of
The key to Cramer’s Rule is replacing the variable column of interest with the constant column and calculating the determinants. We can then express
and
as a quotient of two determinants.
Cramer’s rule for 2×2 systems
Cramer’s Rule is a method that uses determinants to solve systems of equations that have the same number of equations as variables.
Consider a system of two linear equations in two variables.
The solution using Cramer’s Rule is given as
If we are solving for
the
column is replaced with the constant column. If we are solving for
the
column is replaced with the constant column.
Finding the determinant of a 2×2 matrix is straightforward, but finding the determinant of a 3×3 matrix is more complicated. One method is to augment the 3×3 matrix with a repetition of the first two columns, giving a 3×5 matrix. Then we calculate the sum of the products of entries
down each of the three diagonals (upper left to lower right), and subtract the products of entries
up each of the three diagonals (lower left to upper right). This is more easily understood with a visual and an example.
Find the
determinant of the 3×3 matrix.
Augment
with the first two columns.
From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.
From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.
Questions & Answers
Discuss the differences between taste and flavor, including how other sensory inputs contribute to our perception of flavor.
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include:
1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body
the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature