DLA Terminology and notation

Section 8.2 Terminology and notation

${(i, j)}^{th}$ minor of a square matrix $A$: the determinant of the smaller square matrix obtained from $A$ by removing the $i^{th}$ row and the $j^{th}$ column
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— written $M_{i j}$
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${(i, j)}^{th}$ cofactor of a square matrix $A$: equal to either the corresponding minor of $A$ or its negative, depending on whether $i + j$ is even or odd
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— written $C_{i j}$
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cofactor expansion along the $i^{th}$ row of square matrix $A$: the formula $a_{i 1} C_{i 1} + a_{i 2} C_{i 2} + \dots + a_{i n} C_{i n},$ where $C_{i j}$ denotes the ${(i, j)}^{th}$ cofactor of $A$
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cofactor expansion along the $j^{th}$ column of square matrix $A$: the formula $a_{1 j} C_{1 j} + a_{2 j} C_{2 j} + \dots + a_{n j} C_{n j},$ where again $C_{i j}$ denotes the ${(i, j)}^{th}$ cofactor of $A$
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determinant: the common value of all cofactor expansions of a particular square matrix
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— written $det A$
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When computing cofactor expansions, we are often performing determinant calculations inside determinant calculations, and it becomes awkward to have

det

symbols littered throughout our intermediate calculations. So we will also write

| A |

to mean the determinant of a matrix, especially for actual matrices. For example,

\begin{aligned} A & = [\begin{array}{c} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}] & ⟹ & det A & = | \begin{array}{c} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} | . \end{aligned}