Applications of Matrices and Determinants — Formula Sheet (Class 12 Maths)

Every identity below assumes \(A\) (and \(B\)) are \(n\times n\) and, where an inverse appears, non-singular (\(|A|\neq 0\)). Keep this sheet beside you while practising Exercise 1.8.

Inverse and adjoint

• \(A^{-1} = \dfrac{1}{|A|}\operatorname{adj}A\)
• \(A(\operatorname{adj}A) = (\operatorname{adj}A)A = |A|\,I\)
• \(\operatorname{adj}A = |A|\,A^{-1}\)
• \((\operatorname{adj}A)^{-1} = \operatorname{adj}(A^{-1}) = \dfrac{1}{|A|}A\)

Determinant identities

• \(|\operatorname{adj}A| = |A|^{\,n-1}\)
• \(|\operatorname{adj}(\operatorname{adj}A)| = |A|^{\,(n-1)^2}\)
• \(|A^{-1}| = \dfrac{1}{|A|}\)
• \(|AB| = |A|\,|B|\), \(\quad |kA| = k^{\,n}|A|\)

Adjoint identities

• \(\operatorname{adj}(\operatorname{adj}A) = |A|^{\,n-2}A\)
• \(\operatorname{adj}(AB) = (\operatorname{adj}B)(\operatorname{adj}A)\)
• \(\operatorname{adj}(\lambda A) = \lambda^{\,n-1}\operatorname{adj}A\)
• \((\operatorname{adj}A)^{T} = \operatorname{adj}(A^{T})\)

Inverse / transpose algebra

• \((AB)^{-1} = B^{-1}A^{-1}\), \(\quad (ABC)^{-1} = C^{-1}B^{-1}A^{-1}\)
• \((A^{T})^{-1} = (A^{-1})^{T}\), \(\quad (A^{-1})^{-1} = A\)

Orthogonal matrix

• \(AA^{T} = A^{T}A = I \iff A^{-1} = A^{T}\)
• If \(A\) is orthogonal, \(|A| = \pm 1\).

Solving \(AX = B\)

• Matrix inversion: \(X = A^{-1}B\)  (needs \(|A|\neq 0\)).
• Cramer's rule: \(x_i = \dfrac{\Delta_i}{\Delta}\), \(\;\Delta = |A|\neq 0\); \(\Delta_i\) swaps column \(i\) of \(A\) for \(B\).
• Gaussian elimination: reduce \([A\,|\,B]\) to row-echelon form, then back-substitute (works even if \(A\) is singular or rectangular).

Consistency (\(n\) unknowns)

• \(\rho(A) = \rho([A|B]) = n\): unique solution.
• \(\rho(A) = \rho([A|B]) < n\): infinitely many (\((n-\rho)\)-parameter family).
• \(\rho(A) \neq \rho([A|B])\): inconsistent.

Homogeneous \(AX = O\)

• Always has the trivial solution \(X = O\).
• Non-trivial solution exists \(\iff \rho(A) < n\) \(\;(\iff |A| = 0\) for a square system\()\).