Questions: In the following, let (Ti) be an elementary matrix that performs the given row operation. Find (leftTiright). (A stackrel9 R5 rightarrow R5longrightarrow T1 A) (leftT1right=) (square) (8 R5+R3 rightarrow R3) (A quad ldots quad T2 A) (leftT2right=) (square)

In the following, let (Ti) be an elementary matrix that performs the given row operation. Find (leftTiright).

(A stackrel9 R5 rightarrow R5longrightarrow T1 A)

(leftT1right=)

(square)

(8 R5+R3 rightarrow R3)

(A quad ldots quad T2 A)

(leftT2right=)

(square)

Transcript text: In the following, let $T_{i}$ be an elementary matrix that performs the given row operation. Find $\left|T_{i}\right|$. \[ A \stackrel{9 R_{5} \rightarrow R_{5}}{\longrightarrow} T_{1} A \] \[ \left|T_{1}\right|= \] $\square$ \[ \begin{array}{l} 8 R_{5}+R_{3} \rightarrow R_{3} \\ A \quad \ldots \quad T_{2} A \end{array} \] \[ \left|T_{2}\right|= \] $\square$

Solution

Solution Steps

To find the determinant of an elementary matrix $ T_i $ that performs a given row operation, we need to consider the type of row operation:

Scaling a row by a non-zero scalar: The determinant of the elementary matrix is the scalar itself.
Adding a multiple of one row to another: The determinant of the elementary matrix is 1.
Swapping two rows: The determinant of the elementary matrix is -1.

For the given operations:

$9 R_{5} \rightarrow R_{5}$: This is a scaling operation, so the determinant of $ T_1 $ is 9.
$8 R_{5} + R_{3} \rightarrow R_{3}$: This is an addition operation, so the determinant of $ T_2 $ is 1.

Step 1: Determine the Determinant of $ T_1 $

The operation $ 9 R_{5} \rightarrow R_{5} $ is a row scaling operation. The determinant of an elementary matrix that scales a row by a factor of $ c $ is $ c $. Therefore, the determinant of $ T_1 $ is:

\[ \left| T_1 \right| = 9 \]

Step 2: Determine the Determinant of $ T_2 $

The operation $ 8 R_{5} + R_{3} \rightarrow R_{3} $ is a row addition operation. The determinant of an elementary matrix that adds a multiple of one row to another is 1. Therefore, the determinant of $ T_2 $ is:

\[ \left| T_2 \right| = 1 \]