Math010 Chapter 2
Determinants Study Notes
Chapter Contents
2.1 Determinants by Cofactor Expansion 118
2.2 Evaluating Determinants by Row Reduction 126
2.3 Properties of Determinants; Cramer's Rule 133
Introduction
This chapter discusses "determinants" or, more precisely, "determinant functions."
Determinant functions assign a real number, denoted as $f(A)$, to a matrix variable $A$.
Det(A) is the number from the matrix, whereas A is a matrix itself.
Although determinants originated in solving linear equations, they are not commonly used for that purpose in practical applications. Their significance arises from their interconnections within linear algebra and their role in finding matrix inverses.
2.1 Determinants by Cofactor Expansion
Definition of Determinants
The determinant of a square matrix provides essential properties useful for the matrix's computations and structural analysis.
Determinants of 2x2 Matrices
For a 2x2 matrix:
The determinant is defined as:
Inverse formula (for an invertible 2x2 matrix):
Minors and Cofactors
Minor ($M_{ij}$): The determinant of the submatrix formed by deleting the $i^{th}$ row and $j^{th}$ column from $A$.
Cofactor ($C{ij}$): Defined as $C{ij} = (-1)^{i+j} M_{ij}$.
Inductive Definition of Determinants
The determinant of a $1 imes 1$ matrix:
For larger matrices, determinants are computed using minors and cofactors through cofactor expansion recursively.
Example 1: Finding Minors and Cofactors
Matrix:
Minor of $a{11}$:
Cofactor of $a{11}$:
Minor of $a{32}$:
Cofactor Expansion of Determinants
The determinant can be expanded by cofactor expansion along any row or column:
Historical Notes
The term "determinant" was introduced by Carl Friedrich Gauss in 1801.
The term "minor" was coined by James Sylvester in 1850.
Theorem 2.1.1 (Cofactor Expansion)
The method using cofactors gives:
Regardless of which row or column is chosen to expand, the result remains consistent.
Example 3: Using Cofactor Expansion
Find the determinant of:
Solution:
det(A) = 3(-4) - 1(11) + 0 = -1
Example 4: Smart Choice of Row or Column
Choosing rows/columns with zeros simplifies calculation.
Theorem 2.1.2
For an $n imes n$ triangular matrix,
det(A) = a{11} a{22} … a_{nn} for the product of the diagonal entries.
Example 6: Triangular Matrix Determinant
Demonstration that the determinant of a $4 imes 4$ lower triangular matrix simplifies to a product of diagonal entries, using cofactor expansion.
2.2 Evaluating Determinants by Row Reduction
Row Reduction Method
This involves transforming a matrix into row echelon form, allowing for easier evaluation of the determinant.
Theorems outline how elementary row operations affect determinants:
(a) Multiplying a row by a scalar $k$: det(B) = k det(A) det(B) = -det(A) det(B) = det(A) det(kA) = k^n det(A) det(AB) = det(A) det(B) Ax = b through determinants of modified matrices (replacing columns with the constants vector).
Example: Using Cramer’s Rule
Example problem finds the unique solution for specified equations using Cramer’s rule based on matrix determinants.
Equivalence Theorem
Statements about invertibility and consistency of linear systems are equivalent, including determinants being non-zero as a crucial factor to understand invertibility in relation to systems of equations.
These notes have been meticulously structured to allow for a comprehensive understanding of determinants, their calculations, properties, and applications in linear algebra. They can serve as a substantial study resource.
Determinants Study Notes
Chapter Contents
2.1 Determinants by Cofactor Expansion (Evaluating determinants using minors and cofactors)
2.2 Evaluating Determinants by Row Reduction (Effect of elementary row operations)
2.3 Properties of Determinants; Cramer's Rule (Product theorems, invertibility, and solving systems)
Introduction
Definition: A determinant is a function that assigns a specific scalar, $det(A)$ or $|A|$, to every square matrix $A \in M_{n \times n}$.
Conceptual Difference: $A$ represents a transformation or a grid of numbers, whereas $det(A)$ is a single real number that summarizes specific geometric and algebraic properties of that matrix.
Significance: Determinants indicate whether a matrix is invertible, assist in calculating the volume of parallelepipeds in $\mathbb{R}^n$, and are foundational for finding eigenvalues.
2.1 Determinants by Cofactor Expansion
Determinants of 2x2 and 3x3 Matrices
2x2 Case: For $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$, the determinant is $det(A) = ad - bc$.
3x3 Case (Sarrus' Rule): Only for $3 \times 3$, one can sum the products of the diagonals from top-left to bottom-right and subtract the products of the diagonals from top-right to bottom-left.
Minors and Cofactors
Minor ($M_{ij}$): The determinant of the $(n-1) \times (n-1)$ submatrix that remains after removing the $i^{th}$ row and $j^{th}$ column of $A$.
Cofactor ($C{ij}$): The signed minor, calculated as:
C{ij} = (-1)^{i+j} M_{ij}The Checkerboard Pattern: The sign $(-1)^{i+j}$ follows a pattern:
\begin{bmatrix} + & - & + \ - & + & - \ + & - & + \end{bmatrix}
Theorem 2.1.1: Cofactor Expansion
The determinant of an $n \times n$ matrix $A$ can be computed by multiplying the entries in any row (or column) by their corresponding cofactors and adding the resulting products.
Expansion along the $i^{th}$ row: $det(A) = a{i1}C{i1} + a{i2}C{i2} + \dots + a{in}C{in}$
Expansion along the $j^{th}$ column: $det(A) = a{1j}C{1j} + a{2j}C{2j} + \dots + a{nj}C{nj}$
Theorem 2.1.2: Determinant of Triangular Matrices
If $A$ is a triangular matrix (upper, lower, or diagonal), then $det(A)$ is simply the product of the entries on the main diagonal:
det(A) = a{11} a{22} \dots a_{nn}
2.2 Evaluating Determinants by Row Reduction
Row reduction is often more computationally efficient for large matrices (e.g., $n > 3$) than cofactor expansion.
Elementary Row Operations and Determinants
Row Interchange: If $B$ is obtained by swapping two rows of $A$, then $det(B) = -det(A)$.
Row Scaling: If $B$ is obtained by multiplying a single row of $A$ by a scalar $k$, then $det(B) = k \cdot det(A)$.
Row Addition: If $B$ is obtained by adding a multiple of one row to another row of $A$, then $det(B) = det(A)$.
Zero Determinants
A square matrix $A$ has $det(A) = 0$ if:
It has a row or column of all zeros.
Two rows or columns are identical.
One row or column is a scalar multiple of another.
2.3 Properties of Determinants; Cramer's Rule
Key Algebraic Properties
Determinant of a Product: $det(AB) = det(A)det(B)$.
Determinant of a Scalar Multiple: For an $n \times n$ matrix, $det(kA) = k^n det(A)$.
Determinant of a Transpose: $det(A^T) = det(A)$.
Determinant of an Inverse: If $A$ is invertible, $det(A^{-1}) = \frac{1}{det(A)}$.
Cramer's Rule
Used to solve the system $Ax = b$ when $det(A) \neq 0$.
The solution is given by $x1 = \frac{det(A1)}{det(A)}, x2 = \frac{det(A2)}{det(A)}, \dots, xn = \frac{det(An)}{det(A)}$.
$A_i$ is the matrix obtained by replacing the $i^{th}$ column of $A$ with the constant vector $b$.
The Adjoint Matrix and Inverses
The Adjoint (or Adjugate) of $A$, denoted $adj(A)$, is the transpose of the matrix of cofactors of $A$.
Inverse Formula: If $det(A) \neq 0$, then:
A^{-1} = \frac{1}{det(A)} adj(A)
The Invertible Matrix Theorem (Determinant Version)
A square matrix $A$ is invertible if and only if $det(A) \neq 0$.
Chapter Contents
2.1 Determinants by Cofactor Expansion: Utilizing minors and cofactors to break down complex matrices into smaller sub-problems.
2.2 Evaluating Determinants by Row Reduction: Using elementary row operations to transform matrices into triangular forms, which allows for immediate computation.
2.3 Properties of Determinants; Cramer's Rule: Exploring algebraic identities, the relationship between determinants and invertibility, and solving linear systems through specific ratios.
Introduction
Definition of Determinant Function: A determinant is a function $f: M_{n \times n}(\mathbb{R}) \to \mathbb{R}$ that maps a square matrix $A$ to a scalar value. It acts as a measure of the "scaling factor" of the linear transformation described by the matrix.
Geometric Interpretation: In $\mathbb{R}^2$, the absolute value of the determinant of a $2 \times 2$ matrix represents the area of the parallelogram formed by its column vectors. In $\mathbb{R}^3$, it represents the volume of the parallelepiped. A negative determinant signifies a change in orientation.
Significance: While not always efficient for large-scale numerical computation, determinants are theoretically vital for understanding matrix stability, finding eigenvalues, and theoretical proofs regarding vector spaces.
2.1 Determinants by Cofactor Expansion
Determinants of 2x2 and 3x3 Matrices
2x2 Matrix: For $A = \begin{bmatrix} a{11} & a{12} \ a{21} & a{22} \end{bmatrix}$, the determinant is $det(A) = a{11}a{22} - a{12}a{21}$.
3x3 Matrix (Sarrus' Rule): A visual shortcut where you sum products of diagonal entries from left to right and subtract the products of diagonal entries from right to left (Note: this only works for $3 \times 3$ matrices).
Minors and Cofactors
Minor ($M{ij}$): The determinant of the submatrix created by removing the $i^{th}$ row and $j^{th}$ column. It represents the "local" contribution of the entry $a{ij}$.
Cofactor ($C{ij}$): The "signed minor," defined as $C{ij} = (-1)^{i+j} M_{ij}$. The sign $(-1)^{i+j}$ follows a checkerboard pattern starting with $+$ at the $(1,1)$ position:
\begin{bmatrix} + & - & + \ - & + & - \ + & - & + \end{bmatrix}
Theorem 2.1.1: Cofactor Expansion
The determinant can be computed along any row or column. To minimize work, always choose the row or column with the most zeros.
Expansion Formula (along row $i$):
det(A) = a{i1}C{i1} + a{i2}C{i2} + \dots + a{in}C{in}
Theorem 2.1.2: Determinant of Triangular Matrices
If $A$ is an upper triangular, lower triangular, or diagonal matrix, $det(A)$ is the product of its diagonal entries:
det(A) = a{11}a{22} \dots a_{nn}$$
2.2 Evaluating Determinants by Row Reduction
Row reduction is generally more efficient ($O(n^3)$ operations) than cofactor expansion ($O(n!)$ operations) for large matrices.
Elementary Row Operations and their Effects
Row Interchange: Swapping two rows multiplies the determinant by $-1$.
$det(B) = -det(A)$
Row Scaling: Multiplying a row by a scalar $k$ multiplies the determinant by $k$.
$det(B) = k \cdot det(A)$
Row Addition: Adding a multiple of one row to another does not change the determinant.
$det(B) = det(A)$
Zero Determinant Conditions
A matrix $A$ has $det(A) = 0$ if it contains a row of zeros, two identical rows, or if one row is a linear combination of others (linearly dependent rows).
2.3 Properties of Determinants; Cramer's Rule
Fundamental Algebraic Properties
Multiplication: $det(AB) = det(A)det(B)$. Note that $det(A+B) \neq det(A) + det(B)$ generally.
Scalar Multiplication: For $n \times n$ matrix $A$, $det(kA) = k^n det(A)$, because each of the $n$ rows is scaled by $k$.
Transpose: $det(A^T) = det(A)$. This implies that anything true for rows regarding determinants is also true for columns.
Inverses: If $A$ is invertible, $det(A^{-1}) = \frac{1}{det(A)}$.
Adjoint (Adjugate) Matrix and the Inverse
Adjoint ($adj(A)$): The transpose of the matrix of cofactors.
Property: $A \cdot adj(A) = det(A) \cdot I$.
Inverse Formula: If $det(A) \neq 0$, then $A^{-1} = \frac{1}{det(A)} adj(A)$.
Cramer's Rule
Applicable only to square systems $Ax = b$ where $det(A) \neq 0$.
Formula: $xi = \frac{det(Ai)}{det(A)}$, where $A_i$ is the matrix $A$ with column $i$ replaced by the vector $b$.
The Invertible Matrix Theorem (Updated)
For a square matrix $A$, the following are equivalent:
$A$ is invertible.
$Ax = 0$ has only the trivial solution.
The reduced row echelon form of $A$ is $I_n$.
$det(A) \neq 0$.