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Q12SE

Expert-verifiedFound in: Page 165

Book edition
5th

Author(s)
David C. Lay, Steven R. Lay and Judi J. McDonald

Pages
483 pages

ISBN
978-03219822384

**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.**

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.

In this concept, the **parallelogram** contains 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 of *A* have nonzero area.

The **determinant** of *A* is nonzero. It happens if and only if *A* is **invertible**.

Hence, 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.

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