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Chapter 3: Q11E (page 165)

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.

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

Short Answer

Expert verified

The value of the determinant is \( - 18\).

Step by step solution

01

Expand the determinant about the first column

The determinant can be expanded as shown below:

\(\left| {\begin{aligned}{*{20}{c}}3&5&{ - 6}&4\\0&{ - 2}&3&{ - 3}\\0&0&1&5\\0&0&0&3\end{aligned}} \right| = {\left( { - 1} \right)^{1 + 1}} \cdot 3\left| {\begin{aligned}{*{20}{c}}{ - 2}&3&{ - 3}\\0&1&5\\0&0&3\end{aligned}} \right|\)

02

Expand the determinant about the first column

The determinant can be expanded as shown below:

\({\left( { - 1} \right)^{1 + 1}} \cdot 3\left| {\begin{aligned}{*{20}{c}}{ - 2}&3&{ - 3}\\0&1&5\\0&0&3\end{aligned}} \right| = 3{\left( { - 1} \right)^{1 + 1}} \cdot \left( { - 2} \right)\left| {\begin{aligned}{*{20}{c}}1&5\\0&3\end{aligned}} \right|\)

03

Expand the determinant about the first column

The determinant can be expanded as follows:

\(\begin{aligned}{c}3{\left( { - 1} \right)^{1 + 1}} \cdot \left( { - 2} \right)\left| {\begin{aligned}{*{20}{c}}1&5\\0&3\end{aligned}} \right| = \left( { - 6} \right){\left( { - 1} \right)^{1 + 1}}\left( 1 \right)\left( 3 \right)\\ = - 18\end{aligned}\)

So, the value of the determinant is \( - 18\).

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Most popular questions from this chapter

Compute the determinant in Exercise 5 using a cofactor expansion across the first row.

5. \(\left| {\begin{aligned}{*{20}{c}}{\bf{2}}&{\bf{3}}&{ - {\bf{3}}}\\{\bf{4}}&{\bf{0}}&{\bf{3}}\\{\bf{6}}&{\bf{1}}&{\bf{5}}\end{aligned}} \right|\)

Question: 11. Find the area of the parallelogram determined by the points \(\left( {1,4} \right),\)\(\left( { - 1,5} \right),\)\(\left( {3,9} \right),\) and \(\left( {5,8} \right)\). How can you tell that the quadrilateral determined by the points is actually a parallelogram?

Compute the determinants of the elementary matrices given in Exercise 25-30.

27. \(\left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\k&0&1\end{array}} \right]\).

Question: In Exercises 31鈥�36, mention an appropriate theorem in your explanation.

33. Let A and B be square matrices. Show that even thoughABand BAmay not be equal, it is always true that\(det{\rm{ }}AB = det{\rm{ }}BA\).

Is it true that \(det \left( {A + B} \right) = det A + det B\)? Experiment with four pairs of random matrices as in Exercise 44, and make a conjecture.
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