5.2 Properties of Determinants

Determinants are powerful tools in linear algebra, revealing key properties of matrices. They help us understand matrix invertibility, solve systems of equations, and calculate volumes.

The properties of determinants, like multiplicativity and behavior under row operations, simplify complex calculations. These properties connect determinants to matrix transformations, eigenvalues, and other fundamental concepts in linear algebra.

Multiplicative Property of Determinants

Determinant of a Product of Square Matrices

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• The determinant of a product of square matrices equals the product of their determinants: $det(AB) = det(A) \cdot det(B)$
• This property simplifies the calculation of determinants for matrix products
  • Example: If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$, then $det(AB) = det(A) \cdot det(B) = (1 \cdot 4 - 2 \cdot 3) \cdot (5 \cdot 8 - 6 \cdot 7) = -2 \cdot -2 = 4$

Determinant of an Invertible Matrix

• If $A$ is an invertible matrix, then $det(A^{-1}) = 1/det(A)$
  • This property relates the determinants of a matrix and its inverse
  • Example: If $det(A) = 4$, then $det(A^{-1}) = 1/4$

Determinant of a Product of Multiple Square Matrices

• The multiplicative property of determinants extends to the product of any finite number of square matrices: $det(A_1A_2...A_n) = det(A_1) \cdot det(A_2) \cdot ... \cdot det(A_n)$
  • This property allows for the calculation of the determinant of a product of multiple matrices
  • Example: If $A_1 = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, $A_2 = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$, and $A_3 = \begin{bmatrix} 9 & 10 \\ 11 & 12 \end{bmatrix}$, then $det(A_1A_2A_3) = det(A_1) \cdot det(A_2) \cdot det(A_3) = -2 \cdot -2 \cdot -2 = -8$

Row and Column Operations on Determinants

Interchanging Rows or Columns

• Interchanging any two rows or columns of a matrix changes the sign of its determinant
  • This property is useful when simplifying the calculation of determinants
  • Example: If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, then interchanging the rows of $A$ results in $\begin{bmatrix} 3 & 4 \\ 1 & 2 \end{bmatrix}$, and $det(\begin{bmatrix} 3 & 4 \\ 1 & 2 \end{bmatrix}) = -det(A)$

Scalar Multiplication of a Row or Column

• Multiplying a row or column of a matrix by a scalar $k$ multiplies the determinant by $k$
  • This property allows for the simplification of determinants by factoring out common factors
  • Example: If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and the first row is multiplied by 2, the resulting matrix is $\begin{bmatrix} 2 & 4 \\ 3 & 4 \end{bmatrix}$, and $det(\begin{bmatrix} 2 & 4 \\ 3 & 4 \end{bmatrix}) = 2 \cdot det(A)$

Adding a Multiple of a Row or Column to Another

• Adding a multiple of one row or column to another row or column does not change the value of the determinant
  • This property is often used in conjunction with Gaussian elimination to simplify the calculation of determinants
  • Example: If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and the first row is added to the second row, the resulting matrix is $\begin{bmatrix} 1 & 2 \\ 4 & 6 \end{bmatrix}$, and $det(\begin{bmatrix} 1 & 2 \\ 4 & 6 \end{bmatrix}) = det(A)$

Special Cases: Identical or Zero Rows/Columns

• If a matrix has two identical rows or columns, its determinant is zero
  • This property follows from the fact that interchanging identical rows or columns should not change the determinant, but it also changes the sign, leading to the conclusion that the determinant must be zero
  • Example: If $A = \begin{bmatrix} 1 & 2 \\ 1 & 2 \end{bmatrix}$, then $det(A) = 0$
• If a matrix has a row or column consisting entirely of zeros, its determinant is zero
  • This property follows from the fact that the determinant is a linear function of each row or column
  • Example: If $A = \begin{bmatrix} 1 & 2 \\ 0 & 0 \end{bmatrix}$, then $det(A) = 0$

Determinant vs Transpose

Equality of Determinants

• The determinant of a matrix $A$ is equal to the determinant of its transpose $A^T$: $det(A) = det(A^T)$
  • This property allows for the simplification of determinant calculations by choosing to work with either the original matrix or its transpose, whichever is more convenient
  • Example: If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, then $A^T = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}$, and $det(A) = det(A^T) = 1 \cdot 4 - 2 \cdot 3 = -2$

Simplifying Determinant Calculations

• The equality of determinants between a matrix and its transpose can be used to simplify the calculation of determinants
  • In some cases, the transpose of a matrix may have a more convenient structure for calculating the determinant, such as more zeros or easily factorable terms
  • Example: If $A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$, then $A^T = \begin{bmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{bmatrix}$. It may be easier to calculate $det(A^T)$ using cofactor expansion along the first row, as it contains the smallest numbers

Determinant of a Matrix Product

Product of Determinants

• The determinant of a product of matrices is equal to the product of their determinants: $det(AB) = det(A) \cdot det(B)$
  • This property allows for the simplification of determinant calculations by breaking down a matrix product into its constituent matrices
  • Example: If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$, then $det(AB) = det(A) \cdot det(B) = (1 \cdot 4 - 2 \cdot 3) \cdot (5 \cdot 8 - 6 \cdot 7) = -2 \cdot -2 = 4$

Singular Matrix Products

• If either $A$ or $B$ is a singular matrix (i.e., has a determinant of zero), then the product $AB$ is also singular, and $det(AB) = 0$
  • This property follows from the multiplicative property of determinants, as the product of any number with zero is zero
  • Example: If $A = \begin{bmatrix} 1 & 2 \\ 3 & 6 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$, then $det(A) = 0$, and consequently, $det(AB) = det(A) \cdot det(B) = 0 \cdot -2 = 0$

Determinant of a Matrix Raised to a Power

• The determinant of a matrix raised to a power $k$ is equal to the determinant of the matrix raised to the power $k$: $det(A^k) = (det(A))^k$
  • This property follows from the repeated application of the multiplicative property of determinants
  • Example: If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, then $det(A^2) = (det(A))^2 = (-2)^2 = 4$

Non-Commutativity of Matrix Multiplication

• When analyzing the determinant of a matrix product, the order of the matrices matters, as matrix multiplication is not commutative in general: $AB \neq BA$
  • This property means that the determinant of a matrix product depends on the order of the matrices being multiplied
  • Example: If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$, then $AB = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$ and $BA = \begin{bmatrix} 23 & 34 \\ 31 & 46 \end{bmatrix}$. While $det(AB) = det(BA) = 4$, the matrices $AB$ and $BA$ are not equal