# Linear Algebra Examples

Step 1
The cross product of two vectors and can be written as a determinant with the standard unit vectors from and the elements of the given vectors.
Step 2
Set up the determinant with the given values.
Step 3
Choose the row or column with the most elements. If there are no elements choose any row or column. Multiply every element in row by its cofactor and add.
Step 3.1
Consider the corresponding sign chart.
Step 3.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 3.3
The minor for is the determinant with row and column deleted.
Step 3.4
Multiply element by its cofactor.
Step 3.5
The minor for is the determinant with row and column deleted.
Step 3.6
Multiply element by its cofactor.
Step 3.7
The minor for is the determinant with row and column deleted.
Step 3.8
Multiply element by its cofactor.
Step 3.9
Step 4
Evaluate .
Step 4.1
The determinant of a matrix can be found using the formula .
Step 4.2
Simplify the determinant.
Step 4.2.1
Simplify each term.
Step 4.2.1.1
Multiply by .
Step 4.2.1.2
Multiply .
Step 4.2.1.2.1
Multiply by .
Step 4.2.1.2.2
Multiply by .
Step 4.2.2
Step 5
Evaluate .
Step 5.1
The determinant of a matrix can be found using the formula .
Step 5.2
Simplify the determinant.
Step 5.2.1
Simplify each term.
Step 5.2.1.1
Multiply by .
Step 5.2.1.2
Multiply by .
Step 5.2.2
Subtract from .
Step 6
Evaluate .
Step 6.1
The determinant of a matrix can be found using the formula .
Step 6.2
Simplify the determinant.
Step 6.2.1
Simplify each term.
Step 6.2.1.1
Multiply by .
Step 6.2.1.2
Multiply by .
Step 6.2.2