Inverse Matrices and Matrix Equations Dr Shildneck IDENTITY

Inverse Matrices and Matrix Equations Dr. Shildneck

IDENTITY MATRICES Identity 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 MATRICES Identity Matrices are square with the following characteristics : - 1’s down the diagonal - zero’s every where else 2 x 2 Identity 3 x 3 Identity

INVERSE MATRICES Inverse 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 2 x 2 Matrices To find the inverse of a 2 x 2 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 2 x 2 Matrices Find the inverse of the matrix A. A= a c b d

-1 1 A= 1. 2. 3. 4. a -c b d - Find the determinant : ad - bc Switch the “down” diagonal. Change the signs of the “up” diagonal. Multiply by 1 over the determinant.

Inverses of 2 x 2 Matrices Given the inverse of A is. . . -1 A = 1 Det(A) d -b -c a What happens when the determinant is equal to zero?

Inverses of 2 x 2 Matrices Find the inverse of the matrix A. A= 2 6 1 4

Inverses of 2 x 2 Matrices Find the inverse of the matrix A. B= 3 6 2 4

Matrix Equations Solving 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. 5 x = 30 2. 2 x + 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 side 2. 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 side 2. 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 side 2. Find the inverse of A (multiplier) 3. Multiply by the inverse on the appropriate side (both sides of “=“) 4. Simplify your answer for X.

Multiply The Following

Now, Take that answer and set equal to

So, Now we have…

Which Through Equivalent Matrices GIVES US THE SYSTEM… NOW SOLVE THE SYSTEM…

You Should have gotten the answer… (-2, 2)

Next, Let’s Use Inverse Matrices to Solve the Matrix Equation…

First, Find the Inverse of the multiplier… Determinant = -2 – 15 = -17

Next, Multiply Both Sides by the inverse…

What Do You notice? So, what do you notice about our answers to the system and the matrix equation? What do you think Matrix X was?

Given that This System had the Same Answer As This Matrix Equation… WHAT CAN YOU CONCLUDE ABOUT HOW THE SYSTEM AND MATRIX EQUATION RELATE?

Use Your Conclusion to Write THE FOLLOWING SYSTEM AS A MATRIX EQUATION

Did You Get… COEFFICIENT S Matrix VARIABLES Matrix ARGUMENTS Matrix

Try Writing This System as a Matrix Equation

Systems and Matrices The fact that we can write and Nx. N system as a Matrix Equation allows us to use Inverse Matrices to solve the Matrix Equation rather than multi-step algebraic manipulations. Furthermore, for systems bigger than 2 x 2, this process allows us to quickly solve the system in one step, rather than a page full of steps!

Systems and Matrices The process for solving any system of equations using matrices is as follows: 1. Write the system as a matrix equation AX=B, where A = the coefficient matrix, X = the variable matrix, and B = the argument matrix 2. Find the inverse of A and multiply. 3. The solution to the system is given by X = A-1 B

EXAMPLE 1 Step 1: Write the Matrix Equation. Step 2: Enter Matrix A and Matrix B in the calculator. Step 3: Solve by multiplying A-1 B. Step 4: Write the solution as a set of coordinates.