The Master Formulas
Inverse: $A^{-1} = \frac{1}{|A|} adj(A)$Characteristic Equation: $|A - \lambda I| = 0$ (To find Eigenvalues)
Dimensional Analysis of a Matrix
While a matrix itself is an array, the Determinant of a matrix containing physical units (like a Jacobian in Physics) must have units equal to the product of the units of its diagonal elements. If you are calculating the area of a parallelogram using a 2x2 matrix of coordinates ($L, L$), the determinant will have units of $[L^2]$ (Area).
Variations: Special Matrices
- Symmetric: $A = A^T$ (Mirrored across diagonal).
- Skew-Symmetric: $A = -A^T$ (Main diagonal must be zeros).
- Idempotent: $A^2 = A$.
- Orthogonal: $AA^T = I$ (Rotation matrices).
Shortcuts & Mnemonics
- The 2x2 Inverse Shortcut: Swap the main diagonal elements, change the signs of the off-diagonal elements, and divide by the determinant.
- The Sarrus Rule: A visual way to calculate 3x3 determinants by repeating the first two columns.
- Mnemonic: "RC Cola" — Always multiply Rows by Columns.
Edge Cases
- $|A| = 0$: The matrix is non-invertible. The system of equations either has No Solution or Infinite Solutions.
- Rectangular Matrices: Cannot have a determinant or an inverse. We use "Pseudo-inverses" in advanced data science, but for JEE, we stick to Square Matrices for these operations.
- Trace of a Matrix: The sum of the main diagonal elements ($Tr(A)$). A key property: $Tr(AB) = Tr(BA)$.
