Warm Up

Inverse Matrices

Three main topics today

• Identity Matrix• Determinant • Inverse Matrix

Identity Matrices

An identity matrix is a square matrix that has 1’s along the main diagonal and 0’s everywhere else.

When you multiply a matrix by the identity matrix, you get the original matrix.

1 0 0

0 1 0

0 0 1

1 0

0 1

Determinant

• The determinant of a square matrix is a constant value that doesn’t have much meaning on its own, but finding it allows us to do other things with matrices.

• We will find determinants of 2x2 matrices by hand, and anything bigger on the calculator.

Determinants

• **To find a determinant you must have a SQUARE MATRIX!!**

For the square matrix A = , the determinant is written as detA, or as det and it’s value is det = ad - bc .

Examples!

Find the determinant:

1. 5 7

11 8 5 8 7 11 40 77

37 40 77

2. 3 2

1 5 3 5 2 1 15 2

1715 2

3. 10 2

0 310(-3) - (-2)(0)

-30 + 0 = -30

As stated before, this value of -30 doesn’t really do anything for us on its own, but we will see how we can use this value later.

4.

2 3 8

6 7 1

4 5 9

Finding a 3x3 determinant is a pain to do by hand, we wil use the calculator for these.

• Enter this matrix into your calculator for matrix [A]

• Go back to homescreen• Select Matrix ---Math---det(• Select matrix ----Names----[A]---- Enter.

Inverse Matrix (A-1)

• The product of any square matrix A and its inverse matrix A-1 is equal to the identity matrix I. We can write this as A A-1 = A-1A = I

For the square matrix A =the

inverse is written as A-1 =

For the square matrix A =

54

32

Find the determinant, the inverse matrix A-1, and show that A A-1 = I .

Example

Solution

detA = det

54

32 = (2 5) - (3 4) = 10 - 12 = -2

24

352

1

12

5.15.2A-1 = =

54

32

12

5.15.2

))1(5()5.14()25())5.2(4(

))1(3()5.12()23())5.2(2(

10

01

AA-1 =

Solution Continued

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You try!

Find the determinant, the inverse, and prove the

Inverses of larger matrices (3x3): We will do this on the calculator

• Enter the matrix under edit• Go to home screen• Select Matrix, then select the corresponding letter

to your matrix. Hit enter.• Press the button. Hit enter.

Homework:

Inverses worksheet, all problems.

http://teachers.henrico.k12.va.us/math/hcpsalgebra2/Documents/4-5/4_5HW.pdf