Find the inverse of A = [[2, 1], [5, 3]].

Step 1: Recall the inverse formula for a 2 × 2 matrix.

For A = [[a, b], [c, d]]:

A^(-1) = 1/(ad - bc) [[d, -b], [-c, a]]

The inverse exists only if the determinant is nonzero.

Step 2: Find the determinant.

det(A) = (2)(3) - (1)(5)

det(A) = 6 - 5

det(A) = 1

Since the determinant is not zero, A is invertible.

Step 3: Interchange the diagonal entries.

The diagonal entries 2 and 3 become 3 and 2.

Step 4: Change the signs of the off-diagonal entries.

1 becomes -1, and 5 becomes -5.

This gives:
[[3, -1], [-5, 2]]

Step 5: Multiply by 1/det(A).

Since det(A) = 1:

A^(-1) = [[3, -1], [-5, 2]]

Therefore, the inverse is [[3, -1], [-5, 2]].