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Determinants and Properties

Definition

The determinant is a scalar value that can be computed from the elements of a square matrix. Determinants have important applications in linear algebra, including solving systems of linear equations, finding the inverse of a matrix, and determining whether a matrix is invertible. The determinant of a matrix is denoted as det(A) or \(\lvert A \rvert\), where \( A \) is a square matrix.

For example, for a 2x2 matrix \( A \): [ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} ] The determinant of \( A \) is calculated as: [ \text{det}(A) = ad - bc ]

Properties of Determinants (Statements Only)

  1. Determinant of a Product: The determinant of the product of two matrices is the product of their determinants. [ \text{det}(A \cdot B) = \text{det}(A) \cdot \text{det}(B) ]

  2. Determinant of a Transpose: The determinant of a matrix is equal to the determinant of its transpose. [ \text{det}(A^T) = \text{det}(A) ]

  3. Determinant of an Inverse: The determinant of the inverse of a matrix (if it exists) is the reciprocal of the determinant of the original matrix. [ \text{det}(A^{-1}) = \frac{1}{\text{det}(A)} ]

  4. Determinant of a Scalar Multiple: If a matrix is multiplied by a scalar \( k \), the determinant of the matrix is multiplied by \( k^n \), where \( n \) is the size of the matrix. [ \text{det}(kA) = k^n \cdot \text{det}(A) ] For a 2x2 matrix, \( \text{det}(kA) = k^2 \cdot \text{det}(A) \).

  5. Determinant of an Identity Matrix: The determinant of an identity matrix is always 1. [ \text{det}(I) = 1 ]

  6. Row Interchange: Swapping two rows (or columns) of a matrix changes the sign of the determinant. [ \text{det}(B) = -\text{det}(A) \quad \text{(where \( B \) is obtained by swapping rows of \( A \))} ]

  7. Zero Row or Column: If a matrix has a row or column of all zeros, its determinant is zero. [ \text{det}(A) = 0 ]

  8. Linearity of Determinants: The determinant is linear in each row. This means if you scale one row of the matrix by a scalar, the determinant is scaled by the same scalar.

Examples

Example 1: Determinant of a 2x2 Matrix

Let \( A \) be a 2x2 matrix: [ A = \begin{bmatrix} 3 & 2 \ 1 & 4 \end{bmatrix} ]

The determinant of \( A \) is calculated as: [ \text{det}(A) = (3 \cdot 4) - (2 \cdot 1) = 12 - 2 = 10 ]

Thus, \( \text{det}(A) = 10 \).

Example 2: Determinant of a 3x3 Matrix

Let \( B \) be a 3x3 matrix: [ B = \begin{bmatrix} 1 & 0 & 2 \ -1 & 3 & 1 \ 2 & 1 & 0 \end{bmatrix} ]

The determinant of a 3x3 matrix is calculated using cofactor expansion along the first row: [ \text{det}(B) = 1 \cdot \text{det}\begin{bmatrix} 3 & 1 \ 1 & 0 \end{bmatrix} - 0 \cdot \text{det}\begin{bmatrix} -1 & 1 \ 2 & 0 \end{bmatrix} + 2 \cdot \text{det}\begin{bmatrix} -1 & 3 \ 2 & 1 \end{bmatrix} ]

Calculating the 2x2 minors: [ \text{det}\begin{bmatrix} 3 & 1 \ 1 & 0 \end{bmatrix} = (3 \cdot 0) - (1 \cdot 1) = -1 ] [ \text{det}\begin{bmatrix} -1 & 3 \ 2 & 1 \end{bmatrix} = (-1 \cdot 1) - (3 \cdot 2) = -1 - 6 = -7 ]

Now, substitute these values into the cofactor expansion: [ \text{det}(B) = 1 \cdot (-1) + 0 + 2 \cdot (-7) = -1 + 0 - 14 = -15 ]

Thus, \( \text{det}(B) = -15 \).

Conclusion

Determinants provide important information about matrices, such as whether a matrix is invertible. Understanding the properties of determinants allows us to simplify complex matrix operations, such as solving systems of linear equations, matrix inversion, and understanding matrix behavior under transformation.