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Chapter 3: Q30E (page 165)

Use Theorem 3 (but not Theorem 4) to show that if two rows of a square matrix A are equal, then \(det A = 0\). The same is true for twocolumns. Why?

Short Answer

Expert verified

It is proved thatif two rows or columns of asquare matrixA are equal, then \(\det A = 0\).

Step by step solution

01

State the determinant of the matrix

According totheorem 3,if an interchange operation between any two rows in matrix A gives a new matrix B, \(\det {\rm{ }}B = - \det {\rm{ }}A\).

02

Step 2:Find the determinant of the matrix

An interchange between two rows or columnscan be written as

\(\det {\rm{ }}B = - \det {\rm{ }}A\).

If two rows or columns are the same, then theirinterchange does not change the matrix.Thus thedeterminant also does not change.

\(\det {\rm{ }}B = \det {\rm{ }}A\)

Both the cases,\(\det {\rm{ }}B = - \det {\rm{ }}A\)and\(\det {\rm{ }}B = \det {\rm{ }}A\), are satisfied only when\(\det {\rm{ }}A = 0\).

Hence, it is proved that if two rows or columns of a square matrix A are equal, then \(\det A = 0\).

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Most popular questions from this chapter

In Exercises 24–26, use determinants to decide if the set of vectors is linearly independent.

24. \(\left( {\begin{aligned}{*{20}{c}}4\\6\\2\end{aligned}} \right)\), \(\left( {\begin{aligned}{*{20}{c}}{ - 7}\\0\\7\end{aligned}} \right)\), \(\left( {\begin{aligned}{*{20}{c}}{ - 3}\\{ - 5}\\{ - 2}\end{aligned}} \right)\)

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

31. What is the determinant of an elementary row replacement matrix?

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.

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

Combine the methods of row reduction and cofactor expansion to compute the determinant in Exercise 13.

13. \(\left| {\begin{aligned}{*{20}{c}}{\bf{2}}&{\bf{5}}&{\bf{4}}&{\bf{1}}\\{\bf{4}}&{\bf{7}}&{\bf{6}}&{\bf{2}}\\{\bf{6}}&{ - {\bf{2}}}&{ - {\bf{4}}}&{\bf{0}}\\{ - {\bf{6}}}&{\bf{7}}&{\bf{7}}&{\bf{0}}\end{aligned}} \right|\)

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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