Question

Use properties of determinants to find the value of each determinant if it is known that $\left|\begin{array}{llr}x & y & z \\ u & v & w \\ 1 & 2 & 3\end{array}\right|=4$. $$ \left|\begin{array}{rrr} x & y & z \\ u & v & w \\ 2 & 4 & 6 \end{array}\right| $$

Name: Problem 44, Chapter 10: Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry
Uploaded: 2021-07-18T15:17:21-08:00
Duration: 1 min 22 s
Channel: Yujie Wang
Description: Use properties of determinants to find the value of each determinant if it is known that | x y z u v w 1 2 3 |=4. | x y z u v w 2 4 6 |

   Use properties of determinants to find the value of each determinant if it is known that $\left|\begin{array}{llr}x & y & z \\ u & v & w \\ 1 & 2 & 3\end{array}\right|=4$.
$$
\left|\begin{array}{rrr}
x & y & z \\
u & v & w \\
2 & 4 & 6
\end{array}\right|
$$

Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry

Michael Sullivan 4th Edition

Chapter 10, Problem 44

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Solved by Expert Yujie Wang

07/18/2021

Step 1

The determinant of a matrix can be affected by row operations. One important property of determinants is that if one row of a matrix is multiplied by a scalar, the determinant of the matrix is also multiplied by that scalar. Show more…

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Use properties of determinants to find the value of each determinant if it is known that $\left|\begin{array}{llr}x & y & z \\ u & v & w \\ 1 & 2 & 3\end{array}\right|=4$. $$ \left|\begin{array}{rrr} x & y & z \\ u & v & w \\ 2 & 4 & 6 \end{array}\right| $$

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Key Concepts

Scalar Multiplication of a Row

One of the fundamental properties of determinants is that if a single row (or column) of a matrix is multiplied by a scalar, then the determinant of the matrix is also multiplied by that scalar. This property is crucial in simplifying determinant computations by factoring out common multipliers from a row.

Linearity with Respect to Rows

Determinants are linear in each row, meaning that if a row of a matrix is replaced by a linear combination (or a scalar multiple) of another row, the overall determinant will adjust correspondingly. In this context, if one row of a matrix is replaced by a scalar multiple of its previous form, the determinant is scaled by that same scalar.

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