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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 the determinant in Exercise 4 using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column.

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

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

32. Suppose that A is a square matrix such that \(det{\rm{ }}{A^3} = 0\). Explain why A cannot be invertible.

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.

atr

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.

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

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 Exercise 9, determine the values of the parameter s for which the system has a unique solution, and describe the solution.

9.

\(\begin{array}{c}s{x_{\bf{1}}} + {\bf{2}}s{x_{\bf{2}}} = - {\bf{1}}\\{\bf{3}}{x_{\bf{1}}} + {\bf{6}}s{x_{\bf{2}}} = {\bf{4}}\end{array}\)
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