# 4.4 & 4.5 Notes Remember: Identity Matrices: If the product of two matrices equal the identity matrix then they are inverses.

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4.4 & 4.5 Notes

Remember: Identity Matrices: If the product of two matrices equal the identity matrix then they are inverses.

IDENTITY MATRIX PROOF a = (-3)(1) + (4)(0) = -3 b = (-3)(0) + (4)(1) = 4 c = (-2)(1) + (6)(0) = -2 d = (-2)(0) + (6)(1) = 6

Step 2) Switch a & d Step 3) Change the signs of b&c Step 1) Find determinant A scalar, Put under 1 Step 4) Multiply scalar

FIND THE INVERSE Step 1: Find Determinant A (scalar), put under 1

Step 2: Step 3: SWITCH 5 AND -2 Change signs of 3 and 1 Step 4: Multiply scalar Answer:

1. Find the inverse Step 1 Answer = Steps 2 & 3 Steps 4: Multiply scalar

Solving Matrix Equations 1.Find the inverse of the matrix next to the variable 2.Multiply both sides by the inverse matrix, the inverse must be on the left side when multiplying -Check for the Identity matrix

Step 1: Find the Inverse Matrix First

( Multiply both sides by the inverse matrix on the left ) Step 2 Multiply rows by columns Solution

Multiply both sides by the inverse matrix) Find the inverse first!!!! Solve the Matrix Equation

1.Subtract Matrix from both sides 2.Find the inverse 3.Multiply inverse by both sides (keep it left)

Homework: Read section 4.4 ***Define Identity and Inverse Matrices Pgs. 227-229; 1-3, 14-32e, 54-60e

4.5 Solving systems using matrices.

A system can be written as a single matrix equation. Linear system Matrix equation Matrix A is called the Coefficient matrix. Matrix X is called the Variable matrix Matrix B is called the Constant matrix A X = B

Solving for x and y Step 2: Find the inverse of the Coefficient Matrix and multiply both sides Step 1: Set up the equation in matrix form

Step 2: Finding the Inverse Matrix

Step 2: Multiply both sides by the Inverse… (2,-2)

USE AN INVERSE MATRIX TO SOLVE THE LINEAR SYSTEM. FIRST, BEGIN BY WRITING THE EQUATIONS IN MATRIX FORM. SECOND, YOU MUST NOW FIND THE INVERSE OF THE COEFFICIENT MATRIX.

FIND THE INVERSE OF THE COEFFICIENT MATRIX.

SOLVE THE SYSTEM BY MULTIPLYING BY THE INVERSE x = 1 AND y = 2 OR (1,2)

More Practice 1. 3. 2. 4. (8,4)(4,4) (-1,-5) (44/5, -26/5)

Homework Read section 4.5 ***Matrix of variables and Matrix of constants Pg.233-235; 1-3, 12-18e, 24-30e, 48-62e

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