Worksheet20: Dickson 109
Current Work (editable)
89. Minors. The determinant of order n - 1 obtained by erasing (or covering up) the row and column crossing at a given element of a determinant of order n is called the minor of that element. For example, in the determinant (6') of order 3, the minors of b_{1}, b_{2}, b_{3} are respectively B_{1} = |a_{2} c_{2}| |a_{3} c_{3}|, B_{2} = |a_{1} c_{1}| |a_{3} c_{3}|, B_{3} = |a_{1} c_{1}| |a_{2} c_{2}|. Again, (6') is the minor of d_{4} in the determinant of order 4 given by (7). 90. Expansion According to the Elements of a Row or Column. In |a_{1} b_{1} c_{1}| (6') D = |a_{2} b_{2} c_{2}| |a_{3} b_{3} c_{3}|, denote the minor of any element by the corresponding capital letter, so that b_{1} has the minor B_{1}, b_{3} has the minor B_{3}, etc., as in §89. We shall prove that D = a_{1} A_{1} - b_{1} B_{1} + c_{1} C_{1}, D = a_{1} A_{1} - a_{2} A_{2} + a_{3} A_{3}, D = -a_{2} A_{2} + b_{2} B_{2} - c_{2} C_{2}, D = -b_{1} B_{1} + b_{2} B_{2} - b_{3} B_{3}, D = a_{3} A_{3} - b_{3} B_{3} + c_{3} C_{3}, D = c_{1} C_{1} - c_{2} C_{2} + c_{3} C_{3}. The three relations at the left (or right) are expressed in words by saying that a determinant D of the third order may be expanded according to the elements of the first, second or third row (or column). To obtain the expansion, we multiply each element of the row (or column) by the minor of the element, prefix the proper sign to the product, and add the signed products. The signs are alternately + and -, as in the diagram + - + - + - + - + For example, by expansion according to the second column, |1 4 5| |2 0 3| |3 0 9| = -4 |2 3| |3 9| = -4 × 9 = -36.
Answer Key (non-editable)
\Par{89. Minors.} The determinant of order $n - 1$ obtained by erasing (or covering up) the row and column crossing at a given element of a determinant of order $n$ is called the \emph{minor} of that element. \begin{Remark} For example, in the determinant \Eq{(6')} of order $3$, the minors of $b_{1}$, $b_{2}$, $b_{3}$ are respectively \[ B_{1} = \left|\begin{array}{cc} a_{2} & c_{2} \\ a_{3} & c_{3} \end{array}\right|,\qquad % B_{2} = \left|\begin{array}{cc} a_{1} & c_{1} \\ a_{3} & c_{3} \end{array}\right|,\qquad % B_{3} = \left|\begin{array}{cc} a_{1} & c_{1} \\ a_{2} & c_{2} \end{array}\right|. \] Again, \Eq{(6')} is the minor of $d_{4}$ in the determinant of order $4$ given by \Eq{(7)}. \end{Remark} \Par{90. Expansion According to the Elements of a Row or Column.} In \[ \Tag{(6')} D = \left|\begin{array}{ccc} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array}\right|, \] denote the minor of any element by the corresponding capital letter, so that $b_{1}$ has the minor $B_{1}$, $b_{3}$ has the minor $B_{3}$, etc., as in §89. We shall prove that %[** Attn alignment] \begin{align*} D &= a_{1} A_{1} - b_{1} B_{1} + c_{1} C_{1}, & D &= a_{1} A_{1} - a_{2} A_{2} + a_{3} A_{3}, \\ % D &= -a_{2} A_{2} + b_{2} B_{2} - c_{2} C_{2}, & D &= -b_{1} B_{1} + b_{2} B_{2} - b_{3} B_{3}, \\ % D &= a_{3} A_{3} - b_{3} B_{3} + c_{3} C_{3}, & D &= c_{1} C_{1} - c_{2} C_{2} + c_{3} C_{3}. \end{align*} The three relations at the left (or right) are expressed in words by saying %[** Attn italic ()] that a \begin{Thm}determinant $D$ of the third order may be expanded according to the elements of the first, second or third row (or column)\end{Thm}. To obtain the expansion, we multiply each element of the row (or column) by the minor of the element, prefix the proper sign to the product, and add the signed products. The signs are alternately $+$ and $-$, as in the diagram \[ \begin{array}{ccc} + & - & + \\ - & + & - \\ + & - & + \end{array} \] \begin{Remark} For example, by expansion according to the second column, \[ \left|\begin{array}{ccc} 1 & 4 & 5 \\ 2 & 0 & 3 \\ 3 & 0 & 9 \end{array}\right| = -4 \left|\begin{array}{cc} 2 & 3 \\ 3 & 9 \end{array}\right| = -4 × 9 = -36. \] \end{Remark}%[** Continues?]
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