A comprehensive exploration of determinants, inverse matrices, and their applications in solving systems of equations, with interactive visualizations.
In the first part of this series, we explored the row picture, column picture, and matrix form of a system of equations, along with the elimination method. In this post, we will dive deeper into determinants, inverse matrices, and their applications in solving systems of equations. We’ll also use interactive visualizations to make these concepts more intuitive.
The determinant is a scalar value that provides critical information about a matrix, such as whether it is invertible or the volume scaling factor of a transformation.
For a 2x2 matrix:
\[A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\]The determinant is calculated as:
\[\text{det}(A) = ad - bc\]Consider the matrix:
\[A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\]The determinant is:
\[\text{det}(A) = (1)(4) - (2)(3) = 4 - 6 = -2\]Since the determinant is non-zero, the matrix is invertible.
The inverse of a square matrix ( A ) is a matrix ( A^{-1} ) such that:
\[A A^{-1} = A^{-1} A = I\]Where ( I ) is the identity matrix.
For a 2x2 matrix:
\[A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\]If ( \text{det}(A) \neq 0 ), the inverse is given by:
\[A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\]Using the matrix ( A ) from the previous example:
\[A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\]The determinant is ( -2 ). The inverse is:
\[A^{-1} = \frac{1}{-2} \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} = \begin{bmatrix} -2 & 1 \\ 1.5 & -0.5 \end{bmatrix}\]A system of equations can be written in matrix form as:
\[AX = B\]Where ( A ) is the coefficient matrix, ( X ) is the column vector of variables, and ( B ) is the column vector of constants. If ( A ) is invertible, the solution is:
\[X = A^{-1}B\]Consider the system:
\[\begin{aligned} x + 2y &= 5 \\ 3x + 4y &= 6 \end{aligned}\]In matrix form:
\[\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 6 \end{bmatrix}\]The inverse of ( A ) is:
\[A^{-1} = \begin{bmatrix} -2 & 1 \\ 1.5 & -0.5 \end{bmatrix}\]Multiply ( A^{-1} ) by ( B ):
\[X = A^{-1}B = \begin{bmatrix} -2 & 1 \\ 1.5 & -0.5 \end{bmatrix} \begin{bmatrix} 5 \\ 6 \end{bmatrix} = \begin{bmatrix} -4 \\ 4.5 \end{bmatrix}\]The solution is ( x = -4, y = 4.5 ).
A matrix is invertible if and only if its determinant is non-zero. This property is crucial in solving systems of equations and performing linear transformations.
Determinants can be used to measure how a linear transformation scales or flips the space. For example, a determinant of ( -1 ) indicates a reflection.
Inverse matrices are widely used in fields like computer graphics, physics simulations, and machine learning to solve systems of linear equations efficiently.
In the next part, we will explore eigenvalues, eigenvectors, and their significance in linear transformations, along with more interactive visualizations.