In evaluating the determinant below, about what row or...

Name: Problem 7, Chapter 12: Elementary and Intermediate Algebra
Uploaded: 2020-06-25T23:52:22-08:00
Duration: 1 min 49 s
Channel: Caleb Wood
Description: In evaluating the determinant below, about what row or column was it expanded? | 5 1 -1 8 7 4 9 7 6 |=-1| 8 7 9 7 |-4| 5 1 9 7 |+6| 5 1 8 7 |

In evaluating the determinant below, about what row or column was it expanded? $$ \left|\begin{array}{rrr} 5 & 1 & -1 \\ 8 & 7 & 4 \\ 9 & 7 & 6 \end{array}\right|=-1\left|\begin{array}{ll} 8 & 7 \\ 9 & 7 \end{array}\right|-4\left|\begin{array}{ll} 5 & 1 \\ 9 & 7 \end{array}\right|+6\left|\begin{array}{ll} 5 & 1 \\ 8 & 7 \end{array}\right| $$

In evaluating the determinant below, about what row or column was it expanded?
$$
\left|\begin{array}{rrr}
5 & 1 & -1 \\
8 & 7 & 4 \\
9 & 7 & 6
\end{array}\right|=-1\left|\begin{array}{ll}
8 & 7 \\
9 & 7
\end{array}\right|-4\left|\begin{array}{ll}
5 & 1 \\
9 & 7
\end{array}\right|+6\left|\begin{array}{ll}
5 & 1 \\
8 & 7
\end{array}\right|
$$

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Key Concepts

Determinant Calculation Techniques

Determinants can be calculated by expanding along any row or column. The choice of which row or column to use is often made to simplify computations, such as by selecting a row or column that contains zeros or elements that make the calculation more straightforward.

Minors and Cofactors

A minor is the determinant of the submatrix that remains after removing the row and column of a specific element. The cofactor is formed by applying a sign factor, given by (–1)^(i+j), to the minor. These concepts are essential to the cofactor expansion process, ensuring the correct sign and value are used for each term.

Cofactor Expansion

Cofactor expansion is a method for computing the determinant of a matrix by expanding along a chosen row or column. In this process, each element of the row or column is multiplied by its cofactor, which includes a sign factor based on its position, and the results are summed to obtain the determinant.

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