Finding the Inverse of a 2 ×2 Matrix In Exercises 35-40, use the formula on page 569 to find the

step 1 Now over here in this question we've been provided a square matrix whose entries are given to us

Finding the Inverse of a 2 ×2 Matrix In Exercises 35-40, use...

Finding the Inverse of a $2 \times 2$ Matrix In Exercises $35-40,$ use the formula on page 569 to find the inverse of the $2 \times 2$ matrix (if it exists).
$$\left[ \begin{array}{rr}{-4} & {-6} \\ {2} & {3}\end{array}\right]$$

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Key Concepts

Matrix Inversion

Matrix inversion is the process of finding a matrix that, when multiplied with the original matrix, gives the identity matrix. This concept is fundamental in solving systems of linear equations and in many areas of applied mathematics. Not all matrices are invertible; a matrix must be square and have a non-zero determinant to have an inverse.

Determinant

The determinant is a scalar value that is computed from the elements of a square matrix and provides important information about the matrix, such as whether it is invertible. For a 2x2 matrix, the determinant is calculated as (ad - bc), and if the determinant is zero, the matrix is singular and does not have an inverse.

2x2 Matrix Inversion Formula

The formula for finding the inverse of a 2x2 matrix is a specific case that uses the determinant of the matrix. Given a 2x2 matrix with elements a, b, c, and d, its inverse is calculated by swapping the elements on the main diagonal, changing the signs of the off-diagonal elements, and dividing each term by the determinant. This concise formula is efficient for computations involving small matrices.

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