Problem 58 - College Algebra

Question

Explain how to find the multiplicative inverse for a \(2 \times 2\) invertible matrix.

Step-by-Step Solution

Answer

For a given 2x2 matrix A = \(\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}\), if \(ad-bc ≠ 0\), its inverse Aˉ¹ is given by Aˉ¹ = 1/(ad-bc) * \(\begin{bmatrix} d & -b \\ -c & a \\ \end{bmatrix}\).

Step 1: Identify the given 2 x 2 matrix

For this exercise, let's consider a 2x2 matrix A given by: A = \(\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}\) where a, b, c, and d are constants.

Step 2: Calculate the determinant of the matrix

The determinant 'det(A)' of the matrix A = \(\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}\) is given by: det(A) = ad - bc. We need to check if det(A) is nonzero because only non-singular matrices (determinant is non-zero) are invertible.

Step 3: Compute the adjoint of the matrix

The adjoint of matrix A, denoted as adj(A), is given by transposing the matrix of the cofactors, that is adj(A) = \(\begin{bmatrix} d & -b \\ -c & a \\ \end{bmatrix}\)

Step 4: Calculate the inverse of the matrix

The inverse of A is obtained by dividing the adjoint of the matrix by the determinant 'det(A)', i.e., Aˉ¹ = adj(A)/det(A) = 1/(ad-bc) * \(\begin{bmatrix} d & -b \\ -c & a \\ \end{bmatrix}\). Therefore, Aˉ¹ is the inverse of the matrix A.