Chapter 38 Square Matrix Although a matrix may have any number of rows and columns square matrices have properties that we can use to solve systems of equations A square matrix is one of the form ID: 675080
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Slide1
Use Inverse Matrices to Solve Linear Systems
Chapter 3.8Slide2
Square Matrix
Although a matrix may have any number of rows and columns,
square matrices
have properties that we can use to solve systems of equationsA square matrix is one of the form , where n is a positive whole numberThat is, the number of rows equals the number of columnsLike the real numbers, square matrices have a special matrix called the identity matrixIn the same way that, for the real numbers, , if I is an identity matrix and A is a square matrix of the same dimensions,
Slide3
Square Matrix
Each size of square matrix has its own identity matrix
For
matrices, the identity matrix isFor matrices, the identity matrix is
For any other square matrix, the identity matrix has 1’s in the left-to-right diagonal and 0’s everywhere else
Slide4
Square Matrix
Multiply
Slide5
Inverse Matrix
Recall that, among the real numbers, for every non-zero number
a
there is another number so that Suppose that A and B represent matrices and that I is the
identity matrix
If
, then B is the inverse of A
(or
A
is the inverse of B) Slide6
Example
Suppose that
and
The product
is
The result is the
identity matrix, so that
is the inverse of
W
e write the inverse of a matrix A as , so Slide7
Inverse Matrix
Recall that we use the inverse of a number to solve equations
If
, then multiplying both sides
Slide8
Inverse Matrix
In the same way, the inverse of a matrix can be used to solve a matrix equation
Note that
we must pay attention to the order of the multiplication
Since we multiplied to the left of
, we must also multiply to the left of
Slide9
Calculating the Inverse of a Matrix
Given
, where
are real numbers, the inverse of A is
Note that his can only make sense if
What this means is that not every square matrix has an inverse
Slide10
Example
Find the inverse of
.
Follow the pattern of
. Then,
.
The inverse is
Slide11
Guided Practice
Find the inverse of the matrix.
Slide12
Guided Practice
Find the inverse of the matrix.
;
;
;
Slide13
Solving a Matrix Equation
To solve a matrix equation, we must find an inverse that will cancel the coefficient matrix
If
A, B, and X are matrices such that , then A is the coefficient matrix and to solve for X we use the inverse of ANote that this is similar to finding the inverse (or reciprocal) of real number a in order to solve the equation Once the inverse is found, multiply
Slide14
Solving a Matrix Equation
Example
Solve the matrix equation
for the matrix X.
How do we know that
X
must be a
matrix?
Slide15
Solving a Matrix Equation
First, find the inverse of
A
Slide16
Solving a Matrix Equation
Next, multiply both sides of the matrix equation
on the left
Slide17
Solving a Matrix Equation
Check your answer.
Slide18
Guided Practice
Solve the matrix equation
Slide19
Guided Practice
Solve the matrix equation
The inverse is
Slide20
Guided Practice
Multiply both sides of the equation by the inverse (on the left)
Since the multiplication on the left is
I
, we have
Slide21
Guided Practice
You may use you calculator to perform the final multiplication
Slide22
Inverse of a
Matrix
The inverse of a matrix can be calculated by hand, but the calculation is lengthy, so you will use your calculatorTo do this, enter the element values, then raise the matrix to the powerUse the Math button to display the result with fraction values Slide23
Guided Practice
Use your calculator to find the inverse of the matrix. Check your result by multiplying
.
Slide24
Guided Practice
Use your calculator to find the inverse of the matrix. Check your result by multiplying
.
;
Slide25
Guided Practice
Use your calculator to find the inverse of the matrix. Check your result by multiplying
.
;
Slide26
Guided Practice
Use your calculator to find the inverse of the matrix. Check your result by multiplying
.
;
Slide27
Solve Linear Systems With a Matrix Equation
Multiply the matrices on the left
Slide28
Solve Linear Systems With a Matrix Equation
Multiply the matrices on the left
The result is a system of equations in two variables:
Slide29
Solve Linear Systems With a Matrix Equation
We can solve systems of equations by converting the system into a matrix equation
We can then solve the system in one step by multiplying both sides of the equation by the inverse of the square matrix
Example: Use an inverse matrix to solve the linear system. Slide30
Solve Linear Systems With a Matrix Equation
Use the coefficients of the equations to form a matrix
Multiply this by the
matrix
Slide31
Solve Linear Systems With a Matrix Equation
Multiply this by the
matrix
Set this equal to the
matrix made using the constants
Slide32
Solve Linear Systems With a Matrix Equation
Solve the equation by multiplying both sides by the inverse of the square matrix. You may use your calculator to find the inverse.
Slide33
Solve Linear Systems With a Matrix Equation
Slide34
Guided Practice
Use an inverse matrix to solve the linear system.
Slide35
Guided Practice
Use an inverse matrix to solve the linear system.
;
; infinite solutions
;
Slide36
Guided Practice
Use an inverse matrix to solve the linear system.
Slide37
Exercise 3.8
Handout