Matrices (Math)

Unit 2: Matrices

1. Introduction to Matrices

  • Definition: A matrix is a rectangular array of numbers organized in rows and columns.

  • Purpose: Matrices are used to organize information, making it easier to compute and manipulate data, especially in systems of linear equations.

2. Types and Structure of Matrices

  • Matrix Definition: An m × n matrix has m rows and n columns.

    • Entries: Denoted by 𝑎𝑖𝑗, where i refers to the row, and j refers to the column.

Types of Matrices

  1. Row Matrix: A matrix with only one row.

    • Example: A = [2, 6, -1, 5] (1 x 4 matrix)

  2. Column Matrix: A matrix with only one column.

    • Example: B = [2, 4] (2 x 1 matrix)

  3. Square Matrix: Equal number of rows and columns.

    • Example: A = | 2 | 5 | -2 | | 3 | 1 | 9 | | 4 | 0 | 5 |

  4. Diagonal Matrix: All non-diagonal entries are zero.

  5. Trace of a Matrix: Sum of diagonal elements, e.g. if A = | 2 | -7 | 9 | | 0 | 3 | 2 | | 8 | 9 | 4 | then Trace(A) = 2 + 3 + 4 = 9.

  6. Null/Zero Matrix: All elements are zero.

    • Example: O = | 0 | 0 | 0 | | 0 | 0 | 0 |

  7. Scalar Matrix: Diagonal entries are equal, non-diagonal are zero.

  8. Upper/Lower Triangular Matrices: Non-zero entries are either above (upper) or below (lower) the main diagonal.

3. Algebra of Matrices

  • Operations: Matrices can be added, subtracted, and multiplied (but not divided in the conventional sense).

  • Matrix Equality: Two matrices are equal if they both have the same dimensions and corresponding entries are equal.

  • Addition and Subtraction: Possible only if dimensions match; performed by adding/subtracting corresponding entries.

  • Scalar Multiplication: Each entry is multiplied by the scalar value.

4. Operations on Matrices

Matrix Addition

  • If A and B are both m x n matrices, then A + B = [a𝑖𝑗 + b𝑖𝑗].

Matrix Multiplication

  • To multiply A (m x n) by B (n x p), the number of columns in A must equal the number of rows in B.

  • Inner Product: The multiplication of a row from A with a column from B yields the entry in resultant matrix.

Example of Multiplication

  • If

    • A = | 1 | 3 | | -1| 0 |

    • B = | -1 | 5 | | 2 | 0 | then AB can be computed using inner products of rows and columns.

5. Inverses of Matrices

  • Definition of Inverse: For matrix A, if there exists a matrix A⁻¹ such that AA⁻¹ = A⁻¹A = I (identity matrix), then A is invertible.

  • Finding Inverse for 2 x 2 Matrix: A = | a | b | | c | d | A⁻¹ = (1/(ad - bc)) * | d | -b | | -c | a |

6. Determinants

  • Definition: A determinant provides a scalar value that can infer properties about the matrix (like invertibility).

  • 2 x 2 Determinant:| a | b | | c | d | det(A) = ad - bc.

  • Existence of Inverse: A square matrix is invertible if its determinant is non-zero.

7. Applications of Matrices

  • Systems of Linear Equations can be expressed in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.

  • Use of Cramer's Rule: A method to solve systems of equations using determinants, where determinants indicate the solvability of the system.

  • Cramer’s Rule: For systems in two variables, the solution can be expressed as follows:

    • x = (det(Dx) / det(D))

    • y = (det(Dy) / det(D)) where D is the coefficient matrix.

8. Homework and Practice Problems

  1. Solve the linear equation: x + 2z = 4, and others using inverses of matrices.

  2. Calculate determinants for specific matrices and check invertibility.

  3. Use Cramer's Rule for solving system equations as described in the units.