Matrices
Introduction to Matrices
- A matrix is a rectangular arrangement of numbers.
- These numbers are called the elements or entries of the matrix.
- Rows run horizontally and columns run vertically in a matrix.
- The dimension of a matrix is given in the form 'm x n' (m rows by n columns).
- A square matrix has the same number of rows as it does columns.
- An identity matrix is a square matrix with ones along the main diagonal and zeroes elsewhere.
Matrix Operations
- Matrices can be added or subtracted element by element if they are of the same dimensions.
- Matrix multiplication is possible if the number of columns in the first matrix equals the number of rows in the second.
- In the multiplication of matrices, order matters (AB ≠ BA).
- The transpose of a matrix is obtained by swapping the rows and columns.
- A matrix can be multiplied by a scalar; each element is multiplied by the scalar.
Determinants and Inverses
- The determinant of a matrix is a specific number that can only be calculated for square matrices.
- If the determinant of a matrix is zero, it is said to be singular or non-invertible.
- The inverse of a matrix A is denoted as A^-1, and it satisfies the property that AA^-1 = A^-1A = I, where I is the identity matrix.
- To find the inverse of a matrix, one has to use the formula A^-1 = 1/det(A) adj(A), where adj(A) is the adjugate of A.
Applications of Matrices
- Matrices are used to solve systems of linear equations through Gaussian elimination or Cramer's rule.
- They are used to perform linear transformations including rotation, dilation, reflection and shearing.
- In computer graphics, matrices are used to manipulate and transform 3D figures.
- In probability and statistics, matrices are used in Markov chains to model and predict behaviour over time.