Inverse of a 3 x 3 matrix

Learning Objectives Calculate the determinant of a 3x3 matrix using cofactor expansion. Construct the matrix of minors for a given 3x3 matrix. Determine the matrix of cofactors using the checkerboard pattern of signs. Find the adjugate (or adjoint) of a 3x3 matrix by transposing the cofactor matrix. Calculate the inverse of a 3x3 matrix using the formula A⁻¹ = (1/det(A)) * adj(A). Identify a singular matrix (a matrix with no inverse) by checking its determinant. Verify the inverse by showing that A * A⁻¹ = I, where I is the identity matrix. How does a computer graphics program 'undo' a complex 3D rotation or scaling? 🤔 It uses the inverse of a matrix to reverse the transformation! This tutorial will guide you through the step-by-step process of finding the inve...

TermDefinitionExample Determinant (det(A) or |A|)A unique scalar value calculated from the elements of a square matrix. For a 3x3 matrix, a non-zero determinant indicates that an inverse exists.For A = [[1, 2], [3, 4]], det(A) = (1*4) - (2*3) = 4 - 6 = -2. Minor (M_ij)The determinant of the 2x2 sub-matrix that remains after deleting the i-th row and j-th column of the original 3x3 matrix.For A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]], the minor M₁₁ is the determinant of [[5, 6], [8, 9]], which is (5*9) - (6*8) = 45 - 48 = -3. Cofactor (C_ij)A 'signed' minor. The cofactor is calculated by multiplying the minor M_ij by (-1)^(i+j). This results in a checkerboard pattern of signs.Using the minor M₁₁ = -3 from the previous example, the cofactor C₁₁ = (-1)^(1+1) * M₁₁ = (1) * (-3) = -3. Ma...

Determinant of a 3x3 Matrix For A = [[a, b, c], [d, e, f], [g, h, i]], det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) This is the formula for finding the determinant using cofactor expansion along the first row. It's the first and most critical step in finding the inverse. Cofactor Formula C_ij = (-1)^(i+j) * M_ij This formula determines the sign of each minor to create the cofactors. The exponent (i+j) creates a 'checkerboard' pattern of signs: [[+, -, +], [-, +, -], [+, -, +]]. The Inverse Matrix Formula A⁻¹ = (1 / det(A)) * adj(A) This is the main formula for the inverse. It states that the inverse is the adjugate matrix with every element multiplied by the reciprocal of the determinant. This formula only works if det(A) ≠ 0.