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I've read this already.

The definition of the determinant that is in this text I have is (for a $J \times J$ matrix $A = [A_{ij}]$): $$|A| = \sum_{j=1}^{J}A_{ij}C_{ij}$$ where $C_{ij}$ is the cofactor of the cell $A_{ij}$ of the matrix $A$. For a $1 \times 1$ matrix $A = [a]$, $|A| = a$.

Let's suppose $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}\text{.}$$ The minors $M_{ij}$ are $$\begin{align} M_{11} &= [d] \\ M_{12} &= [c] \\ M_{21} &= [b] \\ M_{22} &= [a]\text{.} \end{align}$$ The appropriate signs for the cofactors are $$\begin{bmatrix} + & - \\ - & + \end{bmatrix}$$ so $$\begin{align} C_{11} &= +|M_{11}| = d \\ C_{12} &= -|M_{12}| = - c \\ C_{21} &= -|M_{21}| = -b \\ C_{22} &= +|M_{22}| = a\text{.} \end{align}$$ Hence, $$|A| = ad-bc-bc+ad = 2(ad-bc)$$ which I know is obviously wrong - this should be just $ad - bc$. What did I do wrong here?

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Clarinetist
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  • When you do cofactor expansion you only need to expand along one row or one column, so you won't be actually using all four cofactors in your final answer – Rob Mar 23 '16 at 15:17
  • @RobBland AH, so the formula I have actually says to expand along a row. Thanks! – Clarinetist Mar 23 '16 at 15:19
  • It should just be $aC_{11} + cC_{21} = a[d] - c[b] = ad - bc$ expanding along the first column, as an example – Rob Mar 23 '16 at 15:19

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Many thanks to Rob Bland.

Expansion is done along the $i$th row using the definition in the question. So, for example, if $i = 1$, $$|A| = A_{11}C_{11}+A_{12}C_{12}=ad-bc$$ as desired.

Expansion may also be done along a column as well, so alternatively, we may define $$|A| = \sum_{i=1}^{J}A_{ij}C_{ij}$$ for expansion along the $j$th column.

If we look at the first column for example, $$|A| = A_{11}C_{11}+A_{21}C_{21}=ad-cb$$ agreeing with $|A|$ from expansion along the first row.

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