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Chapter 3: Q40Q (page 165)

In Exercises 39 and 40, \(A\) is an \(n \times n\) matrix. Mark each statement True or False. Justify each answer.

40.

a. The cofactor expansion of \(\det A\) down a column is equal to the cofactor expansion along a row.

b. The determinant of a triangular matrix is the sum of the entries on the main diagonal.

Short Answer

Expert verified
  1. The given statement is false.
  2. The given statement is false.

Step by step solution

01

Determine whether the given statement is true or false

a)

Theorem 1states that the determinantof an \(n \times n\) matrix can be computed by cofactor expansionacross any row or down any column. Expansion across the \(i{\mathop{\rm th}\nolimits} \) row using the cofactor in \({C_{ij}} = {\left( { - 1} \right)^{i + j}}\det {A_{ij}}\)gives \(\det A = {a_{i1}}{C_{i1}} + {a_{i2}}{C_{i2}} + ... + {a_{in}}{C_{in}}\).

Cofactor expansion down the \(j{\mathop{\rm th}\nolimits} \) column is \[\det A = {a_{1j}}{C_{1j}} + {a_{2j}}{C_{2j}} + ... + {a_{nj}}{C_{nj}}\].

Thus, statement (a) is false.

02

Determine whether the given statement is true or false

b)

If A is a triangular matrix, then according to theorem 2,det A is the product of the entries on its main diagonal.

Thus, statement (b) is false.

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Most popular questions from this chapter

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

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

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.

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

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?

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.

Find the determinant in Exercise 15, where \[\left| {\begin{array}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{\bf{d}}&{\bf{e}}&{\bf{f}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{array}} \right| = {\bf{7}}\].

15. \[\left| {\begin{array}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{\bf{d}}&{\bf{e}}&{\bf{f}}\\{{\bf{3g}}}&{{\bf{3h}}}&{{\bf{3i}}}\end{array}} \right|\]
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