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Chapter 8: Problem 70

Explain how to determine whether the inverse of a \(2 \times 2\) matrix exists. If so, explain how to find the inverse.

Short Answer

Expert verified
The inverse of a \(2 \times 2\) matrix exists if and only if the determinant is not zero. The determinant can be found using the formula \(ad - bc\), and if it isn't zero, the inverse can be found using \[\[d/det, -b/det\],\ [-c/det, a/det\]\].

Step by step solution

01

Understanding Inverse and Determinant

The inverse of a matrix is a matrix such that if it is multiplied by the original matrix, it results in the identity matrix. The determinant of a \(2 \times 2\) matrix is calculated as \(ad - bc\) for a matrix \(\[\[a, b\],\ [c, d\]\]\). The matrix has an inverse if and only if the determinant is not zero.
02

Computing the Determinant

Finding the determinant of the matrix will tell us if the inverse exists. Compute the determinant of the matrix \(\[\[a, b\],\ [c, d\]\]\) by applying the formula \(ad - bc\). If the result is non-zero, the inverse exists.
03

Finding the Inverse

If the inverse exists, it can be found using the formula \(\[\[d/det, -b/det\],\ [-c/det, a/det\]\]\) where 'det' is the determinant calculated in Step 2. Apply this formula to get the inverse matrix.

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