Inverse Matrices and Matrix EquationsDr. Shildneck

Fall, 2015

IDENTITY MATRICESIdentity Matrices work like the number 1.

When you multiply a matrix by its identity, you get the same matrix back.

Identity Matrices ARE COMMUTATIVE!

[A][I] = [I][A] = [A]

IDENTITY MATRICESIdentity Matrices are square with the following characteristics :

- 1’s down the diagonal- zero’s every where else

[1 00 1 ]2x2

Identity [1 0 00 1 00 0 1 ]

3x3 Identity

INVERSE MATRICESInverse Matrices work like reciprocals.

When you multiply a matrix by its inverse, you get the identity matrix.

Inverse Matrices ARE COMMUTATIVE!

[A][A-1] = [I] = [A-1][A]

Inverses of 2x2 Matrices

To find the inverse of a 2x2 Matrix do the following:

1. Find the determinant

2. Switch the “down” diagonal

3. Change the sign of the “up” diagonal

4. Multiply by “1/determinant”

Inverses of 2x2 Matrices

¿Find the inverse of the matrix A.

A=a bc d

¿A =a bc d

1. Find the determinant : ad - bc

2. Switch the “down” diagonal.

3. Change the signs of the “up” diagonal.

4. Multiply by 1 over the determinant.

-

- -1 1

Inverses of 2x2 Matrices

¿Given the inverse of A is...

A-1= a-b

-cd 1

Det(A)

What happens when the determinant is equal to zero?

Inverses of 2x2 Matrices

¿Find the inverse of the matrix A.

A=2 16 4

Inverses of 2x2 Matrices

¿Find the inverse of the matrix A.

B=2 16 4

Inverses of 2x2 Matrices

¿Find the inverse of the matrix A.

C =12 6-3 -1

Using the CalculatorTo find determinants for matrices bigger than 3x3, do the following:

1. Enter a (square) matrix in the calculator

2. To find a determinant :- Go back to the matrix screen- Tab to “MATH”- Choose #1 - det(- Go back to the matrix screen and pick a matrix

Using the CalculatorTo find the inverses of matrices bigger than 2x2 do the following:

1. Enter a (square) matrix in the calculator

2. To find the inverse :- Go back to the matrix screen and pick a matrix- Press the [x-1] button- Press [ENTER]- if you get decimals, press [MATH] [1][ENTER]

Using the CalculatorFind each of the following.

|1 30 3

5 32 1

8 14 2

1 03 9

| [ 1 2 4−1 3 25 1 8 ]

1. 2. The inverse of

[−113

243

−3 2 183

−32−56

]-530

Matrix EquationsSolving Matrix Equations is much like solving linear equations…

1. You want to isolate the unknown matrix by…

2. Adding/Subtracting matrices as needed

3. Getting rid of the matrix multipled with the unknown matrix

Equations…Solve each of the following WITHOUT using DIVISION.

1. 5x = 30 2. 2x + 8 = 24

Equations…Solve each of the following MATRIX equations for X.

1. AX = B 2. AX + C = B

Equations…Solve each of the following MATRIX equations for X.

1. =

1. Add/Subtract to get AX (or XA) on one side2. Find the inverse of A (multiplier)

3. Multiply by the inverse on the appropriate side (both sides of “=“)4. Simplify your answer for X.

Equations…Solve each of the following MATRIX equations for X.

2. =

1. Add/Subtract to get AX (or XA) on one side2. Find the inverse of A (multiplier)

3. Multiply by the inverse on the appropriate side (both sides of “=“)4. Simplify your answer for X.

Equations…Solve each of the following MATRIX equations for X.

1. =

1. Add/Subtract to get AX (or XA) on one side2. Find the inverse of A (multiplier)

3. Multiply by the inverse on the appropriate side (both sides of “=“)4. Simplify your answer for X.

ASSIGNMENTAssignment # 7 – Inverses of 2x2 Matrices and Matrix Equations