Use the formula to find the inverse of matrix
$\text{\hspace{0.17em}}A.\text{\hspace{0.17em}}$ Verify your answer by augmenting with the identity matrix.
Finding the multiplicative inverse of 3×3 matrices
Unfortunately, we do not have a formula similar to the one for a
$\text{\hspace{0.17em}}2\text{}\times \text{}2\text{\hspace{0.17em}}$ matrix to find the inverse of a
$\text{\hspace{0.17em}}3\text{}\times \text{}3\text{\hspace{0.17em}}$ matrix. Instead, we will augment the original matrix with the identity matrix and use
row operations to obtain the inverse.
Given a
$\text{\hspace{0.17em}}3\text{}\times \text{}3\text{\hspace{0.17em}}$ matrix
To begin, we write the
augmented matrix with the identity on the right and
$\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ on the left. Performing elementary
row operations so that the
identity matrix appears on the left, we will obtain the
inverse matrix on the right. We will find the inverse of this matrix in the next example.
Given a
$\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}3\text{\hspace{0.17em}}$ matrix, find the inverse
Write the original matrix augmented with the identity matrix on the right.
Use elementary row operations so that the identity appears on the left.
What is obtained on the right is the inverse of the original matrix.
Use matrix multiplication to show that
$\text{\hspace{0.17em}}A{A}^{\mathrm{-1}}=I\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}{A}^{\mathrm{-1}}A=I.$
Finding the inverse of a 3 × 3 matrix
Given the
$\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}3\text{\hspace{0.17em}}$ matrix
$\text{\hspace{0.17em}}A,\text{\hspace{0.17em}}$ find the inverse.
Augment
$\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ with the identity matrix, and then begin row operations until the identity matrix replaces
$\text{\hspace{0.17em}}A.\text{\hspace{0.17em}}$ The matrix on the right will be the inverse of
$\text{\hspace{0.17em}}A.\text{\hspace{0.17em}}$
Solving a system of linear equations using the inverse of a matrix
Solving a system of linear equations using the inverse of a matrix requires the definition of two new matrices:
$\text{\hspace{0.17em}}X\text{\hspace{0.17em}}$ is the matrix representing the variables of the system, and
$\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ is the matrix representing the constants. Using
matrix multiplication , we may define a system of equations with the same number of equations as variables as
$AX=B$
To solve a system of linear equations using an
inverse matrix , let
$\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ be the
coefficient matrix , let
$\text{\hspace{0.17em}}X\text{\hspace{0.17em}}$ be the variable matrix, and let
$\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ be the constant matrix. Thus, we want to solve a system
$\text{\hspace{0.17em}}AX=B.\text{\hspace{0.17em}}$ For example, look at the following system of equations.
Recall the discussion earlier in this section regarding multiplying a real number by its inverse,
$\text{\hspace{0.17em}}({2}^{\mathrm{-1}})\text{\hspace{0.17em}}2=\left(\frac{1}{2}\right)\text{\hspace{0.17em}}2=1.\text{\hspace{0.17em}}$ To solve a single linear equation
$\text{\hspace{0.17em}}ax=b\text{\hspace{0.17em}}$ for
$\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}$ we would simply multiply both sides of the equation by the multiplicative inverse (reciprocal) of
$\text{\hspace{0.17em}}a.\text{\hspace{0.17em}}$ Thus,
Questions & Answers
The average annual population increase of a pack of wolves is 25.
Period =2π
if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic.
Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation
of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15)
it's standard equation is x^2 + y^2/16 =1
tell my why is it only x^2? why is there no a^2?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations