Matrices and Determinants - BSEB Inter Study Guide
Matrices and Determinants
Questions test matrix algebra, determinant properties, adjoint and inverse, and solving linear systems. Accuracy in signs matters more than speed because one sign error changes the inverse.
Core rules
- For A of order n, |kA|=k^n|A|; |AB|=|A||B|; |Aᵀ|=|A|.
- A⁻¹ exists only if |A| is not 0, and A⁻¹=(1/|A|)adj A; A(adj A)=|A|I.
- For a 2×2 matrix [[a,b],[c,d]], adj=[[d,−b],[−c,a]] and |A|=ad−bc.
- (AB)⁻¹=B⁻¹A⁻¹; (Aᵀ)⁻¹=(A⁻¹)ᵀ.
- A skew-symmetric matrix has zero diagonal and odd order determinant 0.
Worked example
If A=[[1,2],[3,4]], find A⁻¹. |A|=1·4−2·3=4−6=−2. adj A=[[4,−2],[−3,1]]. Therefore A⁻¹=(−1/2)[[4,−2],[−3,1]]=[[−2,1],[3/2,−1/2]].
Traps
- Writing AB=BA; matrix multiplication is not commutative.
- Using adj A without dividing by |A|.
- Swapping rows and columns in the adjoint incorrectly: cofactors are transposed.
- Treating |A+B|=|A|+|B| as valid.
- For singular A, still writing an inverse.