Matrix Operations and Inverse Matrices

Study Notes

Addition: Matrices of the same size can be added element-wise.
Scalar Multiplication: A matrix can be multiplied by a scalar, which multiplies each element by that scalar.

Definition: A matrix A has an inverse A^(-1) if A A^(-1) = I, where I is the identity matrix.
Properties:
- (AB)^(-1) = B^(-1) A^(-1)
- (A^(-1))^(-1) = A
- I^(-1) = I
Finding the Inverse: Inverse of a 2x2 matrix can be found using the formula: A^(-1) = (1/det(A)) * adj(A), where adj(A) is the adjugate matrix.

Definition: The transpose of a matrix A is a matrix A^T, where elements are swapped across the main diagonal.
Properties:
- (A^T)^T = A
- (AB)^T = B^T A^T
- (A^(-1))^T = (A^T)^(-1)

Identity Matrix: A square matrix with all elements on the main diagonal equal to 1, and all other elements equal to 0.
Zero Matrix: A matrix with all elements equal to 0.
Elementary Matrices: Matrices that can be obtained from the identity matrix by a single elementary row operation.
Row Echelon Form: A matrix is in row echelon form if all nonzero rows are above any all-zero rows, and the leading entry of each nonzero row is to the right of the leading entry of the row above it.

Matrices of the same size can be added element-wise.
A matrix can be multiplied by a scalar, which multiplies each element by that scalar.

A matrix A has an inverse A^(-1) if A A^(-1) = I, where I is the identity matrix.
Properties of inverse matrices:
- The inverse of a product of two matrices is the product of their inverses in reverse order.
- The inverse of the inverse of a matrix is the original matrix itself.
- The inverse of the identity matrix is the identity matrix.

The inverse of a 2x2 matrix can be found using the formula: A^(-1) = (1/det(A)) * adj(A), where adj(A) is the adjugate matrix.

The determinant of the product of two matrices is the product of their determinants.
The determinant of the sum of two matrices is not equal to the sum of their determinants, except for special cases.
The determinant of a matrix multiplied by a scalar is equal to the scalar raised to the power of the number of rows (or columns) of the matrix, multiplied by the determinant of the original matrix.
The determinant of the inverse of a matrix is equal to the reciprocal of the determinant of the original matrix.

The transpose of a matrix A is a matrix A^T, where elements are swapped across the main diagonal.
Properties of the transpose:
- The transpose of the transpose of a matrix is the original matrix.
- The transpose of the product of two matrices is the product of their transposes in reverse order.
- The transpose of the inverse of a matrix is equal to the inverse of the transpose of the original matrix.

An identity matrix is a square matrix with all elements on the main diagonal equal to 1, and all other elements equal to 0.
A zero matrix is a matrix with all elements equal to 0.
An elementary matrix is a matrix that can be obtained from the identity matrix by a single elementary row operation.
A matrix is in row echelon form if all nonzero rows are above any all-zero rows, and the leading entry of each nonzero row is to the right of the leading entry of the row above it.