Problem 70 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.