Matrices: Addition, Subtraction, Multiplication, Inverses, and Transformations

These flashcards cover key concepts from the lecture notes on matrices, including matrix operations, dimensions, scalar multiplication, matrix multiplication, inverse matrices, and geometrical transformations.

What is the dimension of a matrix?

The number of rows by the number of columns in the matrix (m x n).

How are matrices added or subtracted?

By adding or subtracting their corresponding elements. Matrices must have the same dimensions to be added or subtracted.

What is scalar multiplication of a matrix?

Multiplying each element of the matrix by a scalar (number).

Under what condition can two matrices A and B be multiplied together?

Only if the number of columns of A equals the number of rows of B.

What is the determinant of a 2x2 matrix M = [[a, b], [c, d]]?

det M = ad - cb

What is a singular matrix?

A matrix with a determinant of 0.

What is the identity matrix I?

A square matrix with 1s on the main diagonal and 0s elsewhere (e.g., [[1, 0], [0, 1]] for a 2x2 matrix).

What is the inverse of a matrix M?

A matrix M⁻¹ such that MM⁻¹ = M⁻¹M = I, where I is the identity matrix.

What is the formula for the inverse of a 2x2 matrix M = [[a, b], [c, d]]?

M⁻¹ = (1/det M) * [[d, -b], [-c, a]], provided det M ≠ 0.

How can inverse matrices be used to solve simultaneous equations?

By expressing the equations in matrix form AX = B, then solving for X as X = A⁻¹B.

What matrix represents a reflection in the line y = x?

[[0, 1], [1, 0]]

What matrix represents a rotation of 90° clockwise about the origin?

[[0, -1], [1, 0]]

What matrix represents an enlargement with scale factor n and centre O?

[[n, 0], [0, n]]

What are invariant points in a transformation?

Points that do not change position after the transformation.