Matrices and Determinants

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Matrices are essential tools in mathematics used to represent linear equations and transformations.

1. Algebra of Matrices

Order of a Matrix

The order (or dimension) of a matrix is given as m × n, where m is the number of rows and n is the number of columns.

Matrix Addition and Subtraction

Two matrices can be added or subtracted if and only if they have the same order. To perform these operations, add or subtract the corresponding elements.

Scalar Multiplication

To multiply a matrix by a scalar (a constant number), multiply every element in the matrix by that scalar.

Matrix Multiplication

To multiply two matrices A and B, the number of columns in A must equal the number of rows in B. The resulting matrix has the rows of A and the columns of B. Multiplication is performed using the 'row-by-column' rule.

2. Determinants

A determinant is a scalar value calculated from a square matrix.

• 2 × 2 Matrix: For A = [[a, b], [c, d]], |A| = (ad - bc).
• 3 × 3 Matrix: Calculated by expanding along any row or column using minors and cofactors. The pattern of signs is (+ - +).

3. Inverse of a 2 × 2 Matrix

The inverse of a matrix A (denoted as A⁻¹) exists only if the determinant is non-zero (non-singular matrix). For A = [[a, b], [c, d]]:

A⁻¹ = (1/|A|) × [[d, -b], [-c, a]]

If |A| = 0, the matrix is singular and has no inverse.