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Which of the following errors would be detected when a trial balance is extracted?
A trial balance checks whether total debits equal total credits. If different amounts are posted on the debit and credit sides of the same transaction, the totals will not agree and the error will be detected. The other errors — complete omissions, reversed entries, and misclassification between accounts — all maintain the arithmetic equality of debits and credits, so they go undetected by a trial balance.
Let $A=\begin{pmatrix}i&-i\\-i&i\end{pmatrix}$ where $i=\sqrt{-1}$. The system $A^8\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}8\\64\end{pmatrix}$ has:
$A = \begin{pmatrix}i&-i\\-i&i\end{pmatrix}$. Note $\det(A) = i^2-i^2=0$, so $A$ is singular.
$A^2$: compute $(A)^2$. Row1·Col1: $i(i)+(-i)(-i)=-1+(-1)=-2i$... Let's compute: $A^2_{11}=i\cdot i+(-i)(-i)=i^2+i^2=-1-1=-2$. So $A^2 = -2A$... check: $A^2 = -2\begin{pmatrix}i&-i\\-i&i\end{pmatrix}$.
Therefore $A^8 = (-2)^4 A^4 = 16(A^2)^2=16\cdot4A^2=64\cdot(-2A)=-128A$.
Wait: $A^2=-2A \Rightarrow A^4=4A^2=-8A \Rightarrow A^8=-8A^4... $ Correctly: $A^8=(-2)^7A = -128A$.
$-128A\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}8\\64\end{pmatrix}$. Since $\det(A)=0$, check consistency. The equations are linearly dependent but RHS $(8,64)$ is not in column space of $A$. Hence no solution.
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