2.5 - Determinants & Multiplicative Inverses of Matrices

2.5 - Determinants & Multiplicative Inverses of Matrices

DETERMINANT • a real number representation of a square matrix. • The determinant of is a number denoted as or det • a matrix with a nonzero determinant is called nonsingular

Second-Order Determinant • The value of det or • is ad - cb.

0(-6) - 8(-2) = 16 8(6) - 7(4) = 20 Examples • Find the value of • Find the value of

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Third-Order Determinant

Find the value of

Option 2 for finding 140 + 0 + 18- 0 - -120 - 126 = 152

The Identity Matrix • a square matrix whose elements in the main diagonal, from upper left to lower right, are 1s, while all other elements are 0s.

Inverse Matrix • the product of a matrix and it’s inverse produces the identity matrix • only for square matrices • The inverse of matrix A would be denoted as A-1

Inverse of a Second-Order Matrix • First, the matrix must be nonsingular! • Then, if the matrix is nonsingular, an inverse exists. • If the detA = 0, then it is singular and no inverse exists.

Inverse of a Second-Order Matrix • If A = and , • then A-1 =

1st - find the det 8(-1) - 3(9) = -35 2nd - find the inverse or Find the inverse of

DAY 2

Let’s use some technology! • it is important that you know how to do all these operations by hand. • matrices bigger than a second order are time consuming and well as multiplying matrices. • your calculators do all of this, but remember you will have a non-calculator section of your test.

are solving systems and matrices in the same chapter? You can use inverse matrices to solve systems of linear equations!

If we rewrite the system • as a product of matrices: Now, if this were a simple linear equation, like 5x = 15, how would you “get rid of” the 5?

First, find the inverse of • Then, multiply both sides by the inverse. (5, 2)

Use inverse matrices to solve

2.5 - Determinants & Multiplicative Inverses of Matrices