Question

Find the inverse of the matrix if it exists. [1/2 1/3] [ 5 4 ]

Answer 1

To find the inverse of a matrix, calculate the determinant of the matrix, if the determinant is non-zero, find the adjoint of the matrix and multiply it by 1/determinant to get the inverse.

Step-by-step explanation:

To find the inverse of a matrix, we follow these steps:

1. Write the given matrix, let's call it matrix A.
2. Find the determinant of matrix A. If the determinant is equal to zero, then the inverse does not exist.
3. Find the adjoint of matrix A by finding the cofactor of each element and then taking its transpose.
4. Multiply the adjoint of matrix A by 1/determinant of matrix A to get the inverse of matrix A.

Let's apply these steps to the given matrix:

Matrix A = [1/2 1/3]
[5 4]

Determinant of matrix A = (1/2 * 4) - (1/3 * 5) = 2/3 - 5/15 = 4/3 - 1 = 1/3.

Since the determinant is not equal to zero, the inverse exists.

Adjoint of matrix A = [4 -1]
[-5 2]

Inverse of matrix A = (1/3) * [4 -1]
[-5 2] = [4/3 -1/3]
[-5/3 2/3]

Therefore, the inverse of the given matrix is [4/3 -1/3]
[-5/3 2/3].