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Chapter 3: Q33E (page 165)

Question: In Exercises 31–36, 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\(det{\rm{ }}AB = det{\rm{ }}BA\).

Short Answer

Expert verified

It is proved that \(\det {\rm{ }}AB = \det {\rm{ }}BA\).

Step by step solution

01

Write the multiplicative property

Based on theorem 6 of multiplicative property, if A andB are square matrices, then the determinant of the product matrix AB is equal to the product of the determinant of A and the determinant of B.

\(\det AB = \left( {\det A} \right)\left( {\det B} \right)\)

02

Prove the statement

From the theorem,

\(\det AB = \left( {\det A} \right)\left( {\det B} \right)\). … (1)

Replace A by B and B by A.

\(\begin{aligned}{}\det BA &= \left( {\det B} \right)\left( {\det A} \right)\\\det BA &= \left( {\det A} \right)\left( {\det B} \right){\rm{ }}...{\rm{ }}\left( 2 \right)\end{aligned}\)

From equations (1) and (2),

\(\det {\rm{ }}AB = \det {\rm{ }}BA\)

Hence, it is proved that \(\det {\rm{ }}AB = \det {\rm{ }}BA\).

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Most popular questions from this chapter

The expansion of a \({\bf{3}} \times {\bf{3}}\) determinant can be remembered by the following device. Write a second type of the first two columns to the right of the matrix, and compute the determinant by multiplying entries on six diagonals.

Add the downward diagonal products and subtract the upward products. Use this method to compute the determinants in Exercises 15-18. Warning: This trick does not generalize in any reasonable way to \({\bf{4}} \times {\bf{4}}\) or larger matrices.

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

Use Exercise 25-28 to answer the questions in Exercises 31 ad 32. Give reasons for your answers.

32. What is the determinant of an elementary scaling matrix with k on the diagonal?

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.

Question: 17. Show that if A is \({\bf{2}} \times {\bf{2}}\), then Theorem 8 gives the same formula for \({A^{ - {\bf{1}}}}\) as that given by theorem 4 in Section 2.2.

Compute the determinant in Exercise 9 by cofactor expansions. At each step, choose a row or column that involves the least amount of computation.

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