Glossary
2x2 Determinant
The scalar value for a 2x2 matrix A = [[a, b], [c, d]], calculated as (ad - bc).
Example:
If a matrix represents a transformation, its 2x2 Determinant tells you how much the area of a shape changes after that transformation.
2x2 Inverse Formula
A specific formula for finding the inverse of a 2x2 matrix, involving swapping diagonal elements, negating off-diagonal elements, and dividing by the determinant.
Example:
To quickly find the inverse of a 2x2 matrix on a no-calculator section, you'd apply the 2x2 Inverse Formula by hand.
3x3 Determinant
A scalar value calculated for a 3x3 matrix using a more complex expansion method, typically found using a calculator in AP Pre-Calculus.
Example:
When working with 3D transformations, the 3x3 Determinant can represent the scaling factor for volume.
Determinant (det(A))
A scalar value calculated from a square matrix that indicates its invertibility and can represent geometric properties like area or volume scaling.
Example:
Calculating the Determinant of a 2x2 matrix formed by two vectors tells you the area of the parallelogram those vectors span.
Identity Matrix (I)
A square matrix with 1s on its main diagonal and 0s elsewhere, which acts as the multiplicative identity for matrices (A * I = A).
Example:
When you multiply any matrix by the Identity Matrix, you get the original matrix back, similar to how multiplying a number by 1 doesn't change it.
Inverse Matrix (A⁻¹)
For a square matrix A, its inverse A⁻¹ is another matrix such that their product (A * A⁻¹ or A⁻¹ * A) results in the identity matrix.
Example:
If matrix A rotates a point, its Inverse Matrix A⁻¹ would rotate the point back to its original position.
Invertibility Condition
A square matrix has an inverse if and only if its determinant is not zero; if the determinant is zero, the matrix is non-invertible (singular).
Example:
Before attempting to find an inverse, always check the Invertibility Condition; if the determinant is zero, the matrix is singular and has no inverse.
Parallel Vectors (related to determinants)
Two vectors are parallel if one is a scalar multiple of the other, which is indicated by a zero determinant when they form a matrix.
Example:
If the Determinant of a matrix formed by two column vectors is zero, it means those vectors are Parallel Vectors and do not span a unique area.