🧚🏽‍♀️Abstract Linear Algebra I Unit 5 – Determinants: Properties and Applications

What Are Determinants?

• Determinants are scalar values associated with square matrices that encode important properties of the matrix
• Denoted as $det(A)$ or $|A|$ where $A$ is a square matrix
• Calculated using a specific formula involving the entries of the matrix
• Determinant of a 2x2 matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is given by $ad-bc$
• Determinant of a 3x3 matrix can be calculated using the Laplace expansion or Sarrus' rule
  • Laplace expansion involves cofactors and minors of the matrix
  • Sarrus' rule uses a mnemonic device to calculate the determinant
• Determinant of an $n \times n$ matrix can be calculated recursively using cofactor expansion along any row or column
• The value of the determinant is independent of the choice of row or column for cofactor expansion
• Determinants have important applications in linear algebra, such as determining the invertibility of a matrix and solving systems of linear equations

Key Properties of Determinants

• Determinant of the identity matrix is always 1, i.e., $det(I_n) = 1$
• Determinant of a matrix is equal to the determinant of its transpose, i.e., $det(A) = det(A^T)$
• If two rows (or columns) of a matrix are interchanged, the determinant changes sign
  • Swapping any two rows (or columns) of a matrix multiplies the determinant by -1
• If a matrix has a row (or column) of zeros, its determinant is 0
• Multiplying a row (or column) of a matrix by a scalar $k$ multiplies the determinant by $k$
  • $det(kA) = k^n det(A)$ where $A$ is an $n \times n$ matrix
• If two rows (or columns) of a matrix are proportional, the determinant is 0
• The determinant of a triangular matrix (upper or lower) is the product of its diagonal entries
• The determinant of a block diagonal matrix is the product of the determinants of its diagonal blocks

Calculating Determinants

• For a 2x2 matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is calculated as $det(A) = ad - bc$
• For a 3x3 matrix, the determinant can be calculated using the Laplace expansion or Sarrus' rule
  • Laplace expansion: $det(A) = a_{11}C_{11} - a_{12}C_{12} + a_{13}C_{13}$ where $C_{ij}$ are the cofactors
  • Sarrus' rule: Multiply the entries along the main diagonal and the two parallel diagonals, then subtract the products of entries along the other three diagonals
• For larger matrices, cofactor expansion can be used recursively along any row or column
  • Cofactor expansion along the $i$-th row: $det(A) = \sum_{j=1}^n (-1)^{i+j} a_{ij} det(M_{ij})$ where $M_{ij}$ is the minor of entry $a_{ij}$
  • Cofactor expansion along the $j$-th column: $det(A) = \sum_{i=1}^n (-1)^{i+j} a_{ij} det(M_{ij})$
• Determinants can also be calculated using row reduction to convert the matrix into an upper triangular form
  • Elementary row operations (except for row swapping) do not change the determinant
  • Once in upper triangular form, the determinant is the product of the diagonal entries

Determinants and Matrix Operations

• The determinant of a product of matrices is the product of their determinants, i.e., $det(AB) = det(A) \cdot det(B)$
  • This property holds for any number of matrices: $det(ABC \cdots) = det(A) \cdot det(B) \cdot det(C) \cdots$
• The determinant of the inverse of a matrix is the reciprocal of the determinant of the original matrix, i.e., $det(A^{-1}) = \frac{1}{det(A)}$
  • This property is only valid for invertible matrices (matrices with non-zero determinants)
• The determinant of a matrix raised to a power is the determinant of the original matrix raised to that power, i.e., $det(A^n) = (det(A))^n$
• If $A$ and $B$ are similar matrices, i.e., $B = P^{-1}AP$ for some invertible matrix $P$, then $det(A) = det(B)$
• The determinant of a matrix is invariant under similarity transformations
• The determinant of a Kronecker product of two matrices is the product of the determinants raised to the power of the size of the other matrix, i.e., $det(A \otimes B) = (det(A))^m \cdot (det(B))^n$ where $A$ is $n \times n$ and $B$ is $m \times m$

Applications in Linear Systems

• A square matrix $A$ is invertible if and only if its determinant is non-zero
  • If $det(A) \neq 0$, then $A$ has a unique inverse $A^{-1}$
  • If $det(A) = 0$, then $A$ is singular (non-invertible) and the linear system $Ax = b$ may have no solution or infinitely many solutions
• Cramer's rule can be used to solve a system of linear equations $Ax = b$ using determinants
  • The solution for the $i$-th variable is given by $x_i = \frac{det(A_i)}{det(A)}$ where $A_i$ is the matrix formed by replacing the $i$-th column of $A$ with the vector $b$
  • Cramer's rule is practical for small systems but becomes computationally expensive for larger ones
• The determinant can be used to find the volume of a parallelepiped spanned by the columns (or rows) of a matrix
  • For a 3x3 matrix $A$, the volume of the parallelepiped spanned by its columns is $|det(A)|$
• Determinants appear in the formulas for the cross product and triple product of vectors in 3D space
  • Cross product: $\vec{a} \times \vec{b} = det(\begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{bmatrix})$
  • Triple product: $\vec{a} \cdot (\vec{b} \times \vec{c}) = det(\begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix})$

Geometric Interpretation

• The determinant of a 2x2 matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ represents the signed area of the parallelogram spanned by the column vectors of $A$
  • The absolute value of the determinant gives the area, while the sign indicates the orientation (clockwise or counterclockwise)
• For a 3x3 matrix, the determinant represents the signed volume of the parallelepiped spanned by the column vectors of the matrix
  • The absolute value of the determinant gives the volume, while the sign indicates the orientation (right-handed or left-handed)
• In higher dimensions, the determinant of an $n \times n$ matrix represents the signed $n$-dimensional volume of the $n$-dimensional parallelepiped spanned by its column vectors
• A matrix with a determinant of 0 corresponds to a transformation that collapses the space onto a lower-dimensional subspace
  • For example, a 3x3 matrix with a determinant of 0 may map a 3D space onto a 2D plane or a 1D line
• The sign of the determinant indicates whether the linear transformation represented by the matrix preserves or reverses the orientation of the space
  • A positive determinant indicates an orientation-preserving transformation (e.g., rotation)
  • A negative determinant indicates an orientation-reversing transformation (e.g., reflection)

Advanced Topics and Theorems

• Cauchy-Binet formula: For matrices $A$ and $B$ of compatible sizes, $det(AB) = \sum_{S} det(A_S) \cdot det(B_S)$ where $S$ ranges over all subsets of size $n$ chosen from $\{1, 2, \ldots, m\}$ and $A_S, B_S$ are the $n \times n$ submatrices of $A$ and $B$ with columns indexed by $S$
• Jacobi's formula: For an invertible matrix $A(t)$ that depends on a parameter $t$, $\frac{d}{dt} det(A(t)) = tr(adj(A(t)) \cdot \frac{dA(t)}{dt})$ where $adj(A)$ is the adjugate matrix of $A$ and $tr$ denotes the trace
• Sylvester's determinant theorem: For matrices $A$ and $B$ of sizes $m \times n$ and $n \times m$ respectively, $det(I_m + AB) = det(I_n + BA)$
• Vandermonde determinant: The determinant of a Vandermonde matrix $V = (a_i^{j-1})_{1 \leq i,j \leq n}$ is given by $det(V) = \prod_{1 \leq i < j \leq n} (a_j - a_i)$
• Determinantal identities: There are various identities involving determinants, such as the Dodgson condensation formula, Plücker relations, and Laplace's expansion for bordered determinants
• Determinants in multilinear algebra: Determinants can be generalized to the context of multilinear algebra, where they are related to the exterior algebra and the notion of orientation of vector spaces
• Determinants in differential geometry: Determinants play a role in the definition of volume forms and the Jacobian determinant, which relates the volume elements of different coordinate systems

Practice Problems and Examples

1. Calculate the determinant of the matrix $A = \begin{bmatrix} 2 & -1 & 0 \\ 3 & 4 & -2 \\ 1 & 0 & 5 \end{bmatrix}$

   • Solution: $det(A) = 2 \cdot (20 - 0) - (-1) \cdot (15 - (-2)) + 0 \cdot (3 - 0) = 40 + 17 + 0 = 57$

2. Determine if the matrix $B = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$ is invertible.

   • Solution: $det(B) = 1 \cdot (45 - 48) - 2 \cdot (28 - 42) + 3 \cdot (32 - 35) = -3 + 28 - 9 = 0$. Since $det(B) = 0$, $B$ is not invertible.

3. Find the area of the parallelogram spanned by the vectors $\vec{u} = (2, 3)$ and $\vec{v} = (1, -1)$.

   • Solution: The area is given by the absolute value of the determinant $\begin{vmatrix} 2 & 1 \\ 3 & -1 \end{vmatrix} = -2 - 3 = -5$. Thus, the area is $|-5| = 5$.

4. Use Cramer's rule to solve the system of linear equations: $2x + 3y = 5$ $x - y = 1$

   • Solution: Let $A = \begin{bmatrix} 2 & 3 \\ 1 & -1 \end{bmatrix}$, $A_x = \begin{bmatrix} 5 & 3 \\ 1 & -1 \end{bmatrix}$, and $A_y = \begin{bmatrix} 2 & 5 \\ 1 & 1 \end{bmatrix}$. Then, $det(A) = -5$, $det(A_x) = -10$, and $det(A_y) = 7$. By Cramer's rule, $x = \frac{det(A_x)}{det(A)} = \frac{-10}{-5} = 2$ and $y = \frac{det(A_y)}{det(A)} = \frac{7}{-5} = -\frac{7}{5}$.

5. Prove that if $A$ is an $n \times n$ matrix and $B$ is obtained from $A$ by adding a multiple of one row to another row, then $det(B) = det(A)$.

   • Proof: Let $A = (a_{ij})$ and suppose $B$ is obtained from $A$ by adding $k$ times the $i$-th row to the $j$-th row. Then, $B = (b_{ij})$ where $b_{ij} = a_{ij}$ for all $i \neq j$ and $b_{jk} = a_{jk} + ka_{ik}$ for all $k$. Using the linearity of the determinant along the $j$-th row, we have: $det(B) = det(a_{j1}, \ldots, a_{jn}) + k \cdot det(a_{i1}, \ldots, a_{in}) = det(A) + k \cdot 0 = det(A)$ since the determinant of a matrix with two identical rows (the $i$-th and $j$-th rows in this case) is 0.