Lesson 12: Determinants & Inverse Matrices

1. Determinant of a Matrix

The determinant is a scalar value computed from a square matrix. It tells us:

• Whether the matrix is invertible (det ≠ 0)
• Volume scaling factor of the transformation
• Orientation change (positive/negative)

For 2×2 matrix:

\[ \det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc \]

Example: det \(\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = 1·4 - 2·3 = 4 - 6 = -2\)

2. Determinant = 0

If \(det(A) = 0\), the matrix is singular (not invertible):

• Columns/rows are linearly dependent
• Transformation squashes space (loses dimension)
• System \(Ax = b\) may have no or infinite solutions

3. Inverse of a Matrix \((2 \times 2)\)

If \(det(A) \not= 0\), A has an inverse \(A^{-1}\) such that \(A A^{-1} = I\).

Formula for \(2×2\):

\[ A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]

Example:\( A = \begin{pmatrix} 4 & 3 \\ 2 & 1 \end{pmatrix},\, det(A) = 4·1 - 3·2 = -2\)

\[ A^{-1} = \frac{1}{-2} \begin{pmatrix} 1 & -3 \\ -2 & 4 \end{pmatrix} = \begin{pmatrix} -1/2 & 3/2 \\ 1 & -2 \end{pmatrix} \]