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The £ terms in (5.14) look unfamiliar. What do they mean? From (5.8), we see that the Laplace expansion of a determinant \A I by its first column can be n expressed in the form £ an |C,, If we replace the first column of \A\ by the i-i column vector d but keep all the other columns intact, then a new determinant will result, which we can call \A{\—the subscript 1 indicating that the first column has been replaced by d. The expansion of \At | by its first column (the d column) will yield the expression £ d:\C:] \, because the elements dt now take the

<=i place of the elements an. Returning to (5.14), we see therefore that

Similarly, if we replace the second column of \A \ by the column vector d, while retaining all the other columns, the expansion of the new determinant \A2\ by its n second column (the d column) will result in the expression d ■ | Ci2 \ ■ When

divided by \A |, this latter sum will give us the solution value x2\ and so on.

This procedure can now be generalized. To find the solution value of the y th variable xy, we can merely replace the / th column of the determinant \A \ by the constant terms J, • • • dn to get a new determinant \Aj\ and then divide \Aj\ by the original determinant |A \. Thus, the solution of the system Ax = d can be expressed as

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