Question

In Problems 43-50, use properties of determinants to find the value of each determinant if it is known that $$ \left|\begin{array}{llr} x & y & z \\ u & v & w \\ 1 & 2 & 3 \end{array}\right|=4 $$ $\left|\begin{array}{ccc}1 & 2 & 3 \\ x-u & y-v & z-w \\ u & v & w\end{array}\right|$

   In Problems 43-50, use properties of determinants to find the value of each determinant if it is known that
$$
\left|\begin{array}{llr}
x & y & z \\
u & v & w \\
1 & 2 & 3
\end{array}\right|=4
$$
 $\left|\begin{array}{ccc}1 & 2 & 3 \\ x-u & y-v & z-w \\ u & v & w\end{array}\right|$

Precalculus: pearson new international edition

Michael Sullivan 9th Edition

Chapter 11, Problem 46

Instant Answer

Step 1

Step 1: Recall the determinant of the original matrix: $$ \left|\begin{array}{ccc} x & y & z \\ u & v & w \\ 1 & 2 & 3 \end{array}\right| = 4 $$ Show more…

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In Problems 43-50, use properties of determinants to find the value of each determinant if it is known that $$ \left|\begin{array}{llr} x & y & z \\ u & v & w \\ 1 & 2 & 3 \end{array}\right|=4 $$ $\left|\begin{array}{ccc}1 & 2 & 3 \\ x-u & y-v & z-w \\ u & v & w\end{array}\right|$

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Key Concepts

Determinant Properties

Determinants are scalar values that are calculated from square matrices and encapsulate important properties of linear transformations, including scaling factors and orientation. One key property is that certain row operations, such as swapping rows, alter the determinant in predictable ways.

Row Interchange Effect

Swapping two rows of a matrix results in multiplying the original determinant by -1. This rule is crucial for understanding how rearrangements of rows, such as reordering the matrix into a different form, directly affect the sign of the determinant while leaving its magnitude unchanged.

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