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Chapter 3: Q12SE (page 165)

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.

Short Answer

Expert verified

The area of a parallelogram is represented by a \(2 \times 2\) matrix A, i.e., its area is given by \(\left| {\det A} \right|\), provided A is invertible.

Step by step solution

01

Use the fact of nonzero area

In this concept, the parallelogramcontains four vectors, which are \(0,{v_1} \ne 0,{v_2} \ne 0,\) and \({v_3} \ne 0\), such that one of \({v_1},{v_2},\) and \({v_3}\) is the sum of the other two vectors if and only if the columns ofA have nonzero area.

02

Use the fact of invertible

The determinantof A is nonzero. It happens if and only if A is invertible.

03

Conclusion

Hence, the area of a parallelogram is represented by a \(2 \times 2\) matrix A i.e., itsarea is given by\(\left| {\det A} \right|\),provided A is invertible.

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Most popular questions from this chapter

Compute \(det{\rm{ }}{B^4}\), where \(B = \left[ {\begin{aligned}{{}{}}1&0&1\\1&1&2\\1&2&1\end{aligned}} \right]\).

In Exercises 21–23, use determinants to find out if the matrix is invertible.

21. \(\left( {\begin{aligned}{*{20}{c}}2&6&0\\1&3&2\\3&9&2\end{aligned}} \right)\)

Let \(u = \left[ {\begin{aligned}{*{20}{c}}a\\b\end{aligned}} \right]\), and \(v = \left[ {\begin{aligned}{*{20}{c}}c\\{\bf{0}}\end{aligned}} \right]\), where a, b, and c are positive integers (for simplicity). Compute the area of the parallelogram determined by u, v, \({\bf{u}} + {\bf{v}}\), and 0, and compute the determinant of \(\left[ {\begin{aligned}{*{20}{c}}{\bf{u}}&{\bf{v}}\end{aligned}} \right]\), and \[\left[ {\begin{aligned}{*{20}{c}}{\bf{v}}&{\bf{u}}\end{aligned}} \right]\]. Draw a picture and explain what you find.

In Exercises 27 and 28, A and B are \[n \times n\] matrices. Mark each statement True or False. Justify each answer.

27. a. A row replacement operation does not affect the determinant of a matrix.

b. The determinant of A is the product of the pivots in any echelon form U of A, multiplied by \({\left( { - {\bf{1}}} \right)^r}\), where r is the number of row interchanges made during row reduction from A to U.

c. If the columns of A are linearly dependent, then \(det\left( A \right) = 0\).

d. \(det\left( {A + B} \right) = det{\rm{ }}A + det{\rm{ }}B\).

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

39.

a. An \(n \times n\) determinant is defined by determinants of \(\left( {n - 1} \right) \times \left( {n - 1} \right)\) submatrices.

b. The \(\left( {i,j} \right)\)-cofactor of a matrix \(A\) is the matrix \({A_{ij}}\) obtained by deleting from A its \(i{\mathop{\rm th}\nolimits} \) row and \[j{\mathop{\rm th}\nolimits} \]column.
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