Q 278

Question

Explain how to decide which row or column you
will use to expand a $3 \times 3$ determinant.

Step-by-Step Solution

Verified

Answer

When we expand $3 \times 3$ determinant by any row or column by expanding minors along the first row, we use the following pattern:

$|\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}| = a_{1} |\begin{matrix} b_{2} & c_{2} \\ b_{3} & c_{3} \end{matrix}| - b_{1} |\begin{matrix} a_{2} & c_{2} \\ a_{3} & c_{3} \end{matrix}| + c_{1} |\begin{matrix} a_{2} & b_{2} \\ a_{3} & b_{3} \end{matrix}|$

Usually we try to pick a row or column that will make the calculation easier.

1Step 1. To explain

To explain how to decide which row or column is used to expand $3 \times 3$ matrix.

2Step 2. Consider an example

Consider a $3 \times 3$ square matrix, $[\begin{matrix} 2 & - 3 & 1 \\ - 3 & 1 & - 2 \\ 1 & 1 & 1 \end{matrix}]$

3Step 3. Evaluating matrix by expanding row or column.

To evaluate a $3 \times 3$ determinant by expanding by minors along the first row, we use the following pattern:

The sign pattern for the minor is as follows:

$|\begin{matrix} + & - & + \\ - & + & - \\ + & - & + \end{matrix}|$

4Step 4. Evaluate the matrix by expanding first row

To evaluate the given matrix by expanding the minors along the first row,

$|\begin{matrix} 2 & - 3 & 1 \\ - 3 & 1 & - 2 \\ 1 & 1 & 1 \end{matrix}| = 2 |\begin{matrix} 1 & - 2 \\ 1 & 1 \end{matrix}| - (- 3) |\begin{matrix} - 3 & - 2 \\ 1 & 1 \end{matrix}| + 1 |\begin{matrix} - 3 & 1 \\ 1 & 1 \end{matrix}|$