įinally, we can put these into a matrix to form the cofactor matrix. Them by 1 or − 1, according to their position in the following matrix:įor instance, 𝐶 is in position ( 1, 2 ), which has a negative sign, Recall that the cofactors can be obtained from the corresponding minors by multiplying Repeating this process, we get nine different determinants, where each time we have removed the row and column that theĬorresponding entry belongs to. The full method will only be needed later, let us begin with the first step in the process, which is the calculation of Method for calculating the determinant of a 3 × 3 matrix using cofactor expansion. As the primary focus of this explainer is 3 × 3 matrices, we will be reviewing the In particular, finding the determinant and the steps involved in doing so are a key component of
Whether a similar approach exists for higher-dimensional cases.Īs we will find out in this explainer, there does exist a formula for the matrix inverse that generalizes the 2 × 2 case. Manipulating the entries of the matrix and dividing by the determinant, provided it is not equal to zero. In the 2 × 2 case, we note that the inverse is obtained by Having said that this extension is possible, it is easier said than done to derive formulas for such matrices or to know
Where 𝐼 is the 𝑛 × 𝑛 identity matrix. 𝐴 ) is an 𝑛 × 𝑛 matrix that satisfies
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