Inverse of a 3x3 Matrix using Adjoint | Don't Memories

To find the inverse of a 3x3 matrix A, use the formula: A⁻¹ = (1/det(A)) * adj(A), where det(A) is the determinant and adj(A) is the adjoin of A. The key condition for this formula to apply is that det(A) must not be zero; if it equals zero, then an inverse does not exist. Therefore, calculating the determinant of matrix A is essential before proceeding with finding its inverse.

To find the inverse of a matrix, start by calculating its determinant using the first row as a reference. Multiply each element in the row with their corresponding determinants from submatrices, applying signs based on position. The calculations yield values that combine to give a final determinant of minus 6. Since this value is not zero, it confirms that the matrix is non-singular and an inverse exists.

To find the adjoint of a matrix A, start by calculating the cofactor matrix for each element. Once you have this cofactor matrix, take its transpose to obtain the adjoint.

To find the co-factors of a matrix, start with the first element, 2. Its co-factor is calculated as (-1)^(1+1) times its minor, resulting in -2 after determining the determinant of remaining elements. For -1, its co-factor equals (-1)^(3) multiplied by 0 minus (0 times 2), yielding a value of +2. The process continues for all elements; ultimately leading to six additional co-factors that need calculation.

To find the inverse of a matrix, first calculate its adjoint and determinant. The formula for the inverse is 1 over the determinant multiplied by the adjoint. For example, with a determinant of -6, multiplying each element of the adjoint by -1/6 yields specific values: 1/3 for certain elements and zero for others. This method provides an efficient way to compute A's inverse compared to elementary operations.