Question

If the inverse exists, find the inv [17,13],[4,3]⁻¹

Answer 1

The inverse of the given matrix A = [[17, 13], [4, 3]] is A⁻¹ = [[3, -13], [-4, 17]].

Step-by-step explanation:

Matrix inversion involves finding a matrix that, when multiplied with the original matrix, results in the identity matrix. The inverse of a 2x2 matrix A = [[a, b], [c, d]] is given by the formula:

`\[ A^(-1) = (1)/(ad - bc) \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]`

For the given matrix A = [[17, 13], [4, 3]], the determinant (ad - bc) is calculated as (17 * 3 - 13 * 4) = 51 - 52 = -1. Since the determinant is not zero, the matrix is invertible. Applying the formula, we obtain:

`\[ A^(-1) = (1)/(-1) \begin{bmatrix} 3 & -13 \\ -4 & 17 \end{bmatrix} = \begin{bmatrix} -3 & 13 \\ 4 & -17 \end{bmatrix} \]`

Thus, the inverse of the given matrix is A⁻¹ = [[3, -13], [-4, 17]].

In practical terms, finding the inverse of a matrix is essential in various applications, such as solving systems of linear equations and transformations. The determinant being non-zero signifies that the matrix is non-singular and has a unique inverse. The provided inverse matrix, when multiplied with the original matrix, will yield the identity matrix, confirming the accuracy of the inversion.