More on Matrices Quiz : Friday, Oct. 16 Unit 1 Test: Oct. 23 ( tentative ) WOTD: affable: adj: courteous and pleasant, sociable, easy to speak to

Identity Matrix a + = a a * = a Matrix I = 2 X 2 Matrix I = 3 x 3 The Identity Matrix must be a Square Matrix 0 1

Inverse of a Matrix a + = 0 a * = 1 -a If A and B are inverses, then... AB = I and BA = I

A determinant tells us whether or not a matrix has an inverse. This will be important when we solve systems of equations using matrices. A matrix has an inverse if and only if the determinant does NOT equal 0.

Determinant of a 2x2 det (A) = If det(A) ≠0, then A has an inverse. Does B have an Inverse? Find determinant of B. Yes… det(B) = -2 ad – bc

Do A and B have inverses?

To find the inverse of a 2 x 2 det(A) = –2 so it has an inverse. Find the inverse of A.

Determinants: Diagonal Method To find the determinant of a 3 x 3 matrix, first recopy the first two columns. Then do 6 diagonal products. The determinant of the matrix is the sum of the bottom products minus the sum of the top products – 168 – ( – 45) – ( – 4) = 118 – 119 = – 1

Find the determinant of this 3x3

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

Ex. 1 Write as a matrix equation.

Ex. 2 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. 3 Solve using matrices. x = -7 y = -4 A B A -1 X = A -1 B AX = B (-7, -4) X

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

Things you should know how to do w/o the calculator 2x23x3det inverse solve a system

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