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Chapter 3: Q3E (page 165)

Each equation in Exercises 1-4 illustrates a property of determinants. State the property

\(\left| {\begin{array}{*{20}{c}}{\bf{3}}&{ - {\bf{6}}}&{\bf{9}}\\{\bf{3}}&{\bf{5}}&{ - {\bf{5}}}\\{\bf{1}}&{\bf{3}}&{\bf{3}}\end{array}} \right| = {\bf{3}}\left| {\begin{array}{*{20}{c}}{\bf{1}}&{ - {\bf{2}}}&{\bf{3}}\\{\bf{3}}&{\bf{5}}&{ - {\bf{5}}}\\{\bf{1}}&{\bf{3}}&{\bf{3}}\end{array}} \right|\)

Short Answer

Expert verified

The property used is one row of matrix A is multiplied by a scalar k to produce matrix B.

Step by step solution

01

Recall the property of the determinant

If one row of the determinant is multiplied by k to produce B, then \(\det B = k\det A\).

02

Apply the property for the given determinant

For the given determinant, 3 is the multiple of row 1.

So, the property used is one row of matrix A is multiplied by a scalar k to produce matrix B.

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Most popular questions from this chapter

Question: 12. Use the concept of area of a parallelogram to write a statement about a \(2 \times 2\) matrix A that is true if and only if A is invertible.

Compute the determinants in Exercises 9-14 by cofactor expnasions. At each step, choose a row or column that involves the least amount of computation.

\(\left| {\begin{array}{*{20}{c}}{\bf{4}}&{\bf{0}}&{ - {\bf{7}}}&{\bf{3}}&{ - {\bf{5}}}\\{\bf{0}}&{\bf{0}}&{\bf{2}}&{\bf{0}}&{\bf{0}}\\{\bf{7}}&{\bf{3}}&{ - {\bf{6}}}&{\bf{4}}&{ - {\bf{8}}}\\{\bf{5}}&{\bf{0}}&{\bf{5}}&{\bf{2}}&{ - {\bf{3}}}\\{\bf{0}}&{\bf{0}}&{\bf{9}}&{ - {\bf{1}}}&{\bf{2}}\end{array}} \right|\)

Find the determinants in Exercises 5-10 by row reduction to echelon form.

\(\left| {\begin{aligned}{*{20}{c}}{\bf{3}}&{\bf{3}}&{ - {\bf{3}}}\\{\bf{3}}&{\bf{4}}&{ - {\bf{4}}}\\{\bf{2}}&{ - {\bf{3}}}&{ - {\bf{5}}}\end{aligned}} \right|\)

Let \(u = \left[ {\begin{array}{*{20}{c}}3\\0\end{array}} \right]\), and \(v = \left[ {\begin{array}{*{20}{c}}1\\2\end{array}} \right]\). Compute the area of the parallelogram

determined by u, v, \({\bf{u}} + {\bf{v}}\), and 0, and compute the determinant of \(\left[ {\begin{array}{*{20}{c}}{\bf{u}}&{\bf{v}}\end{array}} \right]\). How do they compare? Replace the first entry of v by an arbitrary number x, and repeat the problem. Draw a picture and explain what you find.

In Exercise 19-24, explore the effect of an elementary row operation on the determinant of a matrix. In each case, state the row operation and describe how it affects the determinant.

\(\left[ {\begin{aligned}{*{20}{c}}a&b\\c&d\end{aligned}} \right],\left[ {\begin{aligned}{*{20}{c}}c&d\\a&b\end{aligned}} \right]\)
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