A detailed exploration of the 8 fundamental properties of determinants with examples and visualizations.
Determinants are a fundamental concept in linear algebra, providing insights into matrix properties, linear transformations, and system solvability. In this post, we will explore the 8 key properties of determinants, their significance, and examples to illustrate each property.
The determinant of the identity matrix ( I ) is always 1, regardless of its size.
For a 3x3 identity matrix:
\[I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\]The determinant is:
\[\text{det}(I) = 1\]This property reflects that the identity matrix does not scale or distort space.
If two rows (or columns) of a matrix are exchanged, the determinant changes its sign.
Consider the matrix:
\[A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\]The determinant is:
\[\text{det}(A) = (1)(4) - (2)(3) = -2\]If we exchange the rows:
\[A' = \begin{bmatrix} 3 & 4 \\ 1 & 2 \end{bmatrix}\]The determinant becomes:
\[\text{det}(A') = (3)(2) - (4)(1) = 2\]The sign has reversed.
If a row (or column) of a matrix is multiplied by a constant ( k ), the determinant is also multiplied by ( k ). However, adding a multiple of one row to another does not change the determinant.
For the matrix:
\[A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\]If we multiply the first row by 2:
\[A' = \begin{bmatrix} 2 & 4 \\ 3 & 4 \end{bmatrix}\]The determinant becomes:
\[\text{det}(A') = 2 \cdot \text{det}(A) = 2 \cdot (-2) = -4\]If we add 3 times the first row to the second row:
\[A'' = \begin{bmatrix} 1 & 2 \\ 6 & 10 \end{bmatrix}\]The determinant remains unchanged:
\[\text{det}(A'') = \text{det}(A) = -2\]If two rows (or columns) of a matrix are identical, the determinant is 0.
For the matrix:
\[A = \begin{bmatrix} 1 & 2 \\ 1 & 2 \end{bmatrix}\]The determinant is:
\[\text{det}(A) = (1)(2) - (2)(1) = 0\]This property reflects that the matrix is singular and does not represent a valid transformation.
Subtracting a multiple of one row from another (as in Gaussian elimination) does not change the determinant.
For the matrix:
\[A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\]If we subtract 3 times the first row from the second row:
\[A' = \begin{bmatrix} 1 & 2 \\ 0 & -2 \end{bmatrix}\]The determinant remains:
\[\text{det}(A') = \text{det}(A) = -2\]This property is crucial in solving systems of equations.
If a matrix has a row (or column) of zeroes, its determinant is 0.
For the matrix:
\[A = \begin{bmatrix} 1 & 2 \\ 0 & 0 \end{bmatrix}\]The determinant is:
\[\text{det}(A) = (1)(0) - (2)(0) = 0\]This property indicates that the matrix is singular.
For an upper triangular matrix, the determinant is the product of its diagonal elements.
For the matrix:
\[A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix}\]The determinant is:
\[\text{det}(A) = (1)(4)(6) = 24\]This property simplifies determinant calculations for triangular matrices.
If the determinant of a matrix is 0, the matrix is singular and non-invertible.
For the matrix:
\[A = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}\]The determinant is:
\[\text{det}(A) = (1)(4) - (2)(2) = 0\]Since the determinant is 0, ( A ) is singular and does not have an inverse.
Determinants are a powerful tool in linear algebra, providing insights into matrix properties, transformations, and system solvability. In the next post, we will explore eigenvalues, eigenvectors, and their applications in understanding linear transformations.