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,
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387