If P = $\begin{pmatrix} 2 & -3 \\ 1 & 1 \end{pmatrix}$ , what is P$^-1$

Analyze the question: We need the inverse of a 2×2 matrix P.

Concept: For a matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the inverse is

(swap a and d, change the signs of b and c, divide by the determinant).

Solution:

Here a = 2, b = -3, c = 1, d = 1.

Determinant: ad - bc = (2)(1) - (-3)(1) = 2 + 3 = 5

Adjoint: swap the diagonal and change the signs of the off-diagonal entries:

So

Check: Multiply P by P⁻¹:

• Row 1 × Column 1: (2)(1/5) + (-3)(-1/5) = 2/5 + 3/5 = 1 ✓
• Row 1 × Column 2: (2)(3/5) + (-3)(2/5) = 6/5 - 6/5 = 0 ✓
• Row 2 × Column 1: (1)(1/5) + (1)(-1/5) = 0 ✓
• Row 2 × Column 2: (1)(3/5) + (1)(2/5) = 1 ✓

We get the identity matrix, so the answer is confirmed.

Tip: For a 2×2 inverse, remember "swap the diagonal, negate the off-diagonal, divide by the determinant." Always check by multiplying: you should get the identity matrix. If the determinant is 0, there is no inverse.