Do Now: Solve the system of linear equations using matrices (Row Echelon Form)

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1

622

zyx

zyx

yx

Academy Algebra II/Trig12.3: Systems of linear equations: Determinants, 12.4: InversesUnit 3 Test: Thurs, 10/31 (12.1-12.4, 12.7)

Finding the Determinant for a 2 x 2

dc

baGiven

Then bcaddc

ba gives you the determinant.

Find the determinant of the matrix.

16

231.) 2.)

11

22

Cramer's Rule

dc

ba

tc

sa

y

dc

ba

dt

bs

x ,

Cramer’s Rule can be used to solve a system of equations when the det = 0.

Given the system:

tdycx

sbyax

Using Cramer’s Rule, the solution to the system is given by:

Solve the system using Cramer’s Rule.

136

423

yx

yx1.) 2.)

02

243

yx

x

Finding the Determinant of a 3x3

)()(

det

idbhfageccdhbfgaei

hg

ed

ba

ihg

fed

cba

ihg

fed

cba

Evaluate.

328

264

103

Solve the system using Cramer’s Rule.

432

342

32

zyx

zyx

zyx

12.4: Inverse Matrices

dc

baA

Two matrices are inverses of each other if their product = identity matrix.

The inverse of a 2x2 matrix is

ac

bd

AA

11

Note: Matrix A will not have an inverse ifthe determinant = 0.

Find the inverse of the matrix, if possible.

16

231.) 2.)

32

64

Finding the Inverse for a 3 x 3You will find inverses for a 3 x 3 matrix on the calculator.

Input the following matrix by creating a new matrix or overwriting a current matrix.

Note: Once you are entering #’s in the cells for the matrix you can resize the matrix by selecting the Util menu (F6) – option 6.

1

340

431

011

Finding the Inverse for a 3 x 3

You will find inverses for a 3 x 3 matrixon the calculator.

Input the following matrix by creating a new matrix or overwriting a current matrix.Press Home.Type the name of your matrix and raise it to the -1 exponent. Press ENTER.

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340

431

011

Finding the determinant on calculator

You can also find determinants for a matrixon the calculator.

To find the determinant for this matrix – on your home screen complete the following:Go to MATH menu (Press 2nd 5)Select Matrix, select det(Enter the name of your matrix, close parenthesis. Press ENTER.

340

431

011

Using inverse matrices to solve a linear system.

fdycx

ebyax

Given system:

f

e

y

x

dc

ba

BAX

f

e

dc

ba

y

x

BAX1

1

Solve the system using Inverses.

5

82

yx

yx

Solve the system using Inverses.

532

12

0

yx

zy

zyx