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To evaluate the determinant of a 3 × 3 matrix we choose any row or column of the matrix - this will contain three elements. We then find three products by multiplying each element in the row or column we have chosen by its cofactor. Finally, we sum these three products to find the value of the determinant.
The determinant of a matrix is equal to the determinant of the transposed matrix i.e. Therefore, if there is a zero in the first column, transpose and calculate the determinant or use the alternative formula.
This leaflet will show you how to find the determinant of a 3 × 3 matrix. We will choose the third column with elements 3, 1, 0. The determinant is then given by. Note that we don’t need to work out the cofactor of 0 since it is going to be multiplied by zero. 5 2 = 35 + 4 = 39. The place sign of 3 is. 7 +, so the cofactor is 39. −1 4 − 2 = 26.
We can use these ten properties to find a formula for the determinant of a 2 by 2 matrix: = 0 + ad + (−cb) + 0 = ad − bc. By applying property 3 to separate the individual entries of each row we could get a formula for any other square matrix. However, for a 3 by 3 matrix we’ll have to add the determinants of twenty seven different matrices!
the terms in the determinant of a 3 × 3 matrix is given in Figure 3.4.1. By taking the product of the elements joined by each arrow and attaching the indicated sign to the result, we obtain the six terms in the determinant of the 3 × 3 matrix A =[aij]. Note that this technique for obtaining the terms in a 3×3 determinant does not generalize to
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Given a square matrix A,the determinant of A will be defined as a scalar, to be denoted by det(A) or |A|. We define determinant inductively. That means, we first define determinant of 1×1 and 2×2 matrices. Use this to define determinant of 3×3 matrices. Then, use this to define determinant of 4×4 matrices and so.
