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Inverting 3x3 Matrices

Inverting 3x3 Matrices. Consider the matrix . First we need to find the cofactors . These are found by crossing out the row/column of an entry and calculating the 2x2 determinant created, and then finding the correct sign. For instance A 1 the cofactor of a 1 is. Inverting 3x3 Matrices.

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Inverting 3x3 Matrices

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  1. Inverting 3x3 Matrices Consider the matrix First we need to find the cofactors. These are found by crossing out the row/column of an entry and calculating the 2x2 determinant created, and then finding the correct sign. For instance A1 the cofactor of a1 is

  2. Inverting 3x3 Matrices The signs used are in a chess board pattern: Thus A2 the cofactor of a2 is Etc.

  3. Determinant of a 3x3 Matrix The determinant can now be defined in terms of the cofactors as: det = a1A1+b1B1+c1C1 or det = a1A1+a2A2+a3A3 det = a2A2+b2B2+c2C2 or det = b1B1+b2B2+b3B3 det = a3A3+b3B3+c3C3 or det = c1C1+c2C2+c3C3

  4. Alien Cofactors Note that if you calculate the product of entries with a different row/column then instead of getting the determinant you get zero: a1A2+b1B2+c1C2 = 0or c1A1+c2A2+c3A3 = 0 This result is called the property of alien cofactors (Dr Who beware!) Why is this the case? Basically because you have made two rows or two columns the same, and thus the determinant has to be zero as there will no longer be a unique solution.

  5. Finding the inverse • Find the matrix of the cofactors • Transpose the matrix, by swapping the rows and columns: • Divide by the determinant

  6. Finding the inverse The method on the previous slide is very prone to errors, so check your result. Do this by evaluating the product It should be This is also one way to calculate the determinant.

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