Question

In Exercises 23–30, use expansion by minors to evaluate each determinant. ﻿∣111222−34−5∣\begin{vmatrix}
1 & 1 & 1 \
2 & 2 & 2 \
-3 & 4 & -5
\end{vmatrix}​12−3​124​12−5​​

Accepted Answer

Identify the matrix for which you need to find the determinant: 
$$\begin{bmatrix} 1 & 1 & 1 \ 2 & 2 & 2 \ -3 & 4 & -5 \end{bmatrix}$$ Choose a row or column to expand along. Typically, choose the row or column with the most zeros or simplest numbers to make calculations easier. Here, the first row is a good choice: 
$$	ext{Expand along the first row: } 1, 1, 1$$ Calculate the minors for each element in the first row. For each element, cross out the row and column containing that element and find the determinant of the resulting 2x2 matrix. For example, for the element in the first row and first column (which is 1), the minor is:
$$\begin{vmatrix} 2 & 2 \ 4 & -5 \end{vmatrix}$$ Calculate the cofactors by applying the sign pattern based on the position of each element. The sign pattern for a 3x3 matrix is:
$$\begin{bmatrix} + & - & + \ - & + & - \ + & - & + \end{bmatrix}$$
Multiply each minor by its corresponding sign and the element from the first row. Sum the products of the elements, their cofactors, and minors to get the determinant:
$$	ext{Determinant} = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}$$
where $$a_{ij}$$ are elements of the first row and $$C_{ij}$$ are their cofactors.

In Exercises 23–30, use expansion by minors to evaluate each determinant. $\begin{vmatrix}\)1 & 1 & 1 \\2 & 2 & 2 \\-3 & 4 & -5\(\end{vmatrix}$

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Key Concepts

Determinant of a Matrix

Expansion by Minors

Properties of Determinants

In Exercises 23–30, use expansion by minors to evaluate each determinant. ∣111222−34−5∣\(\begin{vmatrix}\)1 & 1 & 1 \\2 & 2 & 2 \\-3 & 4 & -5\(\end{vmatrix}\)12−312412−5