Quant · Algebra

Matrices and Determinants

2 sub-topics · 24 lessons · 63 questions · 15 PYQs

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Matrices10 lessons · 29 questions · 4 PYQs

Determinants14 lessons · 34 questions · 11 PYQs

Formula sheet

When matrices behave like numbers
$(A+B)^{2}=A^{2}+2AB+B^{2}\iff AB=BA$
Symmetric plus skew-symmetric split
$A=\tfrac12(A+A^{T})+\tfrac12(A-A^{T})$
Nilpotent matrices
$A^{2}=O\Rightarrow(I+A)^{n}=I+nA,\ \det(I+kA)=1$
Inverse from a polynomial relation
$A^{2}-pA+qI=O,\ q\neq0\Rightarrow A^{-1}=\tfrac1q(pI-A)$
Determinant of the adjoint
$|\operatorname{adj}A|=|A|^{n-1},\qquad A\operatorname{adj}A=|A|I$
Matrix multiplication
$(AB)_{ij}=\sum_{k}a_{ik}b_{kj};\quad (m\times n)(n\times p)=m\times p$
Circulant determinant
$\begin{vmatrix}a&b&c\\b&c&a\\c&a&b\end{vmatrix}=-(a^{3}+b^{3}+c^{3}-3abc)$
Expansion of a 3 by 3 determinant
$\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)$