Determinant

222122

11

22

11 or det bababa

ba

ba

ba

The numerical value of a square array of numbers that can be used to solve systems of equations with matrices.

Second-Order Determinant(of a 2 x 2 Matrix)

The word with the matrix

symbol

The Determinant symbol

222122

11

22

11 or det bababa

ba

ba

ba

(Criss-Cross & multiply then subtract)

Example 1

204768 204768 204768 204768

nth-Order Determinant

The determinant of any n x n (square) matrix

Third-Order Determinant(of a 3 x 3 Matrix)

Determinant of what’s leftDeterminant of what’s leftDeterminant of what’s leftSigns alternate, beginning with minus

Then simplify

Example 2

34)3)(1( 34)3)(1(

Select any row & column, then calculate the determinant with

expansion of the minors

34)3)(1(

Identity Matrix for Multiplication

A square matrix, that when multiplied with another square matrix, results in a matrix with no change.The square matrix always consists of 1’s on the diagonal beginning with the first element; the remaining elements are zeros.

100

010

001

10

01

Inverse Matrix (A-1)

A matrix that when multiplied by another matrix results in the identity matrix.

IAA 1

Note: Not all matrices have an inverse – If the determinant of the original matrix has a value of

zero, A-1 does not exist

Example 3

Determine if a matrix exists (Det ≠ 0)

423

-4

Switch places

Same place

Switch Signs

The inverse is used to solve systems of equations with matrices.

11

32

53112

Calculate the determinant:

Determine the inverse:

Multiply each side of the matrix equation by the inverse & solve:

Example 5

)(085.06.10. yxyx

Define the variables:

Let x = amount of 10% bondLet y = amount of 6% bond

Two variables, two equations:x + y = 10,500

Simplify; no decimals.

Write matrix equation: Determine the inverse:

53

111(-5) – 3(1) = -8

3x – 5y = 0

HW: Page 102