# Determinants, Inverse Matrices & Solving

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30-Dec-2015Category

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### Transcript of Determinants, Inverse Matrices & Solving

Determinants, Inverse Matrices & Solving

Notice the different symbol:the straight lines tell you to find the determinant!!(3 * 4)-(-5 * 2)12 - (-10) 22=Finding Determinants of Matrices==

= [(2)(-2)(2) + (0)(5)(-1) + (3)(1)(4)] [(3)(-2)(-1) + (2)(5)(4) + (0)(1)(2)][-8 + 0 +12] --[6 + 40 + 0]4 6 - 40Finding Determinants of Matrices=== -42

Identity matrix:Square matrix with 1s on the diagonal and zeros everywhere else2 x 2 identity matrix 3 x 3 identity matrix The identity matrix is to matrix multiplication as ___ is to regular multiplication!!!!1

Multiply: So, the identity matrix multiplied by any matrix lets the any matrix keep its identity!Mathematically, IA = A and AI = A !!

Inverse Matrix:2 x 2 In words:Take the original matrix. Switch a and d. Change the signs of b and c. Multiply the new matrix by 1 over the determinant of the original matrix.

=Example: Find the inverse of A.

Find the inverse matrix.Det A = 8(2) (-5)(-3) = 16 15 = 1 Matrix A

What happens when you multiply a matrix by its inverse?1st: What happens when you multiply a number by its inverse?A & B are inverses. Multiply them.

Why do we need to know all this?To Solve Problems!Solve for Matrix X.We need to undo the coefficient matrix.Multiply it by its INVERSE!

You can take a system of equations and write it with matrices!!!3x + 2y = 112x + y = 8becomes=Coefficient matrixVariable matrixAnswer matrix

Let A be the coefficient matrix.Multiply both sides of the equation by the inverse of A.=====

Wow!!!!3x + 2y = 112x + y = 8x = 5; y = -23(5) + 2(-2) = 112(5) + (-2) = 8It works!!!!Check:

You TrySolve: 4x + 6y = 142x 5y = -9

You TrySolve: 2x + 3y + z = -13x + 3y + z = 12x + 4y + z = -2

Real Life Example:You have $10,000 to invest. You want to invest the money in a stock mutual fund, a bond mutual fund, and a money market fund. The expected annual returns for these funds are given in the table.You want your investment to obtain an overall annual return of 8%. A financial planner recommends that you invest the same amount in stocks as in bonds and the money market combined. How much should you invest in each fund?

To isolate the variable matrix, RIGHT multiply by the inverse of ASolution: ( 5000, 2500, 2500)

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