University:
The University Of Western Ontario
Course:
CALC 1000A | Calculus I
Academic year:

2020
Views:

237

Pages:

3
Author:

Kyuwon J.

Determinants and Matrix Properties 1. Determinants The determinant of a square matrix A is a scalar value that provides important information about the matrix, such as whether it is invertible. For a 2 × 2 matrix: a b A= , det(A) = ad − bc c d For a 3 × 3 matrix:   a b c A = d e f  , g h i det(A) = a(ei − f h) − b(di − f g) + c(dh − eg) Properties of Determinants • Row or Column Swap: Swapping two rows or columns changes the sign of the determinant. • Proportional Rows or Columns: If two rows or columns are proportional, det(A) = 0. • Triangular Matrix: The determinant of a triangular matrix (upper or lower) is the product of its diagonal elements. • Multiplicative Property: For square matrices A and B: det(AB) = det(A) · det(B) • Invertibility: A matrix A is invertible if and only if det(A) ̸= 0. Applications of Determinants • Determining whether a matrix is invertible. • Calculating the area or volume of geometric shapes using matrices. • Solving systems of linear equations using Cramer’s Rule. 1 2. Matrix Properties 2.1 Types of Matrices • Square Matrix: A matrix with an equal number of rows and columns (n × n). • Diagonal Matrix: A square matrix with non-zero elements only on the diagonal. • Symmetric Matrix: A matrix A where A = AT . • Orthogonal Matrix: A matrix Q where QT Q = I. • Identity Matrix: A square matrix with ones on the diagonal and zeros elsewhere. 2.2 Matrix Operations • Addition and Subtraction: Matrices must have the same dimensions. • Scalar Multiplication: Multiply each element of the matrix by the scalar. • Matrix Multiplication: Defined when the number of columns in the first matrix equals the number of rows in the second. • Transpose: For matrix A, AT is obtained by flipping rows and columns. • Inverse: For A−1 , AA−1 = I, provided det(A) ̸= 0. 2.3 Properties of Matrix Multiplication • Associative: (AB)C = A(BC) • Distributive: A(B + C) = AB + AC • Not Commutative: AB ̸= BA in general. 3. Practice Problems Problem 1: Determinant of a Matrix Compute the determinant of:   1 2 3 A = 4 5 6 7 8 9 Solution: det(A) = 1 5 6 4 6 4 5 −2 +3 8 9 7 9 7 8 2 = 1(5 · 9 − 6 · 8) − 2(4 · 9 − 6 · 7) + 3(4 · 8 − 5 · 7) = 1(−3) − 2(−6) + 3(−3) = −3 + 12 − 9 = 0 Since det(A) = 0, A is not invertible. Problem 2: Properties of Determinants Given:   2 0 0 B = 0 3 0 0 0 4 Find det(B) and interpret. Solution: Since B is a diagonal matrix: det(B) = 2 · 3 · 4 = 24 Interpretation: The determinant indicates that B is invertible, and its rows/columns span a three-dimensional space. Problem 3: Inverse of a Matrix Find the inverse of: 1 2 C= 3 4 Solution: The formula for the inverse of a 2 × 2 matrix is: 1 d −b −1 , where det(C) = ad − bc C = det(C) −c a For C: det(C) = (1)(4) − (2)(3) = 4 − 6 = −2 1 4 −2 −2 1 −1 C = = 1.5 −0.5 −2 −3 1 3