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Expert-verified Found in: Page 165 ### Linear Algebra and its Applications

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.

See the step by step solution

## Step 1: Use the fact of nonzero area

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.

## Step 2: Use the fact of invertible

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

## Step 3: Conclusion

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. ### Want to see more solutions like these? 