Academia Aeternum
तमसो मा ज्योतिर्गमय
Wisdom · Knowledge · Eternity
Order, Operations, Transpose & Invertibility — All Matrix Exercises Solved
5 exercise files · 79 total questions
\((A+B)^T = A^T+B^T;\quad (AB)^T = B^T A^T\)\(A = \tfrac{1}{2}(A+A^T) + \tfrac{1}{2}(A-A^T)\ \text{(sym + skew)}\)\(A A^{-1} = A^{-1}A = I\)\(\text{Matrix mult. defined only if columns of }A = \text{rows of }B\)Step 1 — Before multiplying, always check compatibility of orders (columns of first = rows of second). Step 2 — For symmetric/skew proofs: compute Aᵀ directly and compare entry-by-entry with A or −A. Step 3 — Elementary-operation inverse: apply the same row operation to A and I simultaneously until A becomes I; I becomes A⁻¹. Step 4 — Never assume AB=BA — verify separately if commutativity is claimed.
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