Have we ever seen this phenomenon before? Let’s do some quick multiplication…

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Presentation transcript:

Have we ever seen this phenomenon before? Let’s do some quick multiplication…

What do we call this? 5 __= 5 the (multiplicative) identity 1 Let’s do some quick multiplication…

What will we call this? Identity Matrix I Let’s do some quick multiplication…

Identity Matrix 2x2 Identity Matrix 3x3 Identity Matrix

Identity Matrix AI= A IA= A Any matrix multiplied by the identity matrix will be equal to itself.

Identity Matrix Or

Inverse Matrix A * A -1 = I Any matrix multiplied by its inverse will be the identity matrix.

Solving Systems using Matrices

Matrix Equation A linear system can be written as a matrix equation AX=B Coefficient matrix Variable matrix Answer matrix

Ex. 1 Write as a matrix equation.

Suppose ax = b How do you solve for x? We cannot divide matrices, but we can multiply by the inverse. AX = B A -1 A -1 IX = A -1 B A -1 X = A -1 B Solving Matrix Equations

Ex. 2 Solve using matrices. x = -7 y = -4 A B A -1 X = A -1 B AX = B (-7, -4)

Ex. 3 Solve using matrices x = 5/7 y = 2 (5/7, 2)

Ex. 4 Solve using matrices x = 2 y = -1 z = -2 (2, -1, -2)

Ex. 5 Solve using matrices x = 4 y = -7 z = 2 (4, -7, 2)