Inverse of a Matrix?

📌 CONCEPT: The inverse of a matrix is a special matrix that, when multiplied by the original matrix, gives the identity matrix. The product of a matrix and its inverse is always the identity matrix.

📐 RULE / FORMULA: The formula to find the inverse of a matrix A, if it exists, is A^-1 = (1/det(A)) × adj(A), where det(A) is the determinant of A and adj(A) is the adjoint of A. For 2x2 matrices, this simplifies to A^-1 = (a11a22 - a12a21) / (a11a22 - a12a21) × [[a22, -a12], [-a21, a11]].

💡 WORKED EXAMPLE: Let A = [[2, 3], [4, 5]] and B = [[6, 7], [8, 9]]. We can find the determinants of A and B as 1 and 13, respectively. Then, A^-1 = [[-5, 3], [4, -2]] and B^-1 = [[-1, 7], [4, -6]]. Now, if we compute (AB)^-1, we indeed get B^-1A^-1.

⚠️ COMMON MISTAKE: Students often assume that (AB)^-1 = A^-1B^-1, but this is not correct. The correct formula is (AB)^-1 = B^-1A^-1, which can be verified by multiplying the two matrices together.