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I am currently facing a problem for calculating the unknown in a matrix:

The Determinant is $A=35$

and the matrix is

$$A= \begin{bmatrix} 7 & 8 & 6 & u \\ -5 & 8 & 6 & 7 \\ 4 & 12 & 4 & 4 \\ 3 & 5 & 56 & 7 \\ \end{bmatrix}$$

I want to develop the determinant for the first row and get:

$$35=7* \begin{vmatrix} 8 & 6 & 7\\ 12 & 4 & 4\\ 5 & 56 & 7\\ \end{vmatrix} +5* \begin{vmatrix} 8 & 6 & u\\ 12 & 4 & 4\\ 5 & 56 & 7\\ \end{vmatrix} + 4 * \begin{vmatrix} 8 & 6 & u\\ 8 & 6 & 7\\ 5 & 56 & 7\\ \end{vmatrix} - 3* \begin{vmatrix} 8 & 6 & u\\ 8 & 6 & 7\\ 12 & 4 & 4\\ \end{vmatrix} $$

then I calculate the matrix and get:

$$35 = 7*2612 + 5*(344+672u-20u-1792-504)+4*(336+210+448u-30u-3136-336)-3*(192+504+32u-72u-224-192)$$

as a solution I get:

$$u=4055/5052$$

However, I have 2 questions:

Is it possible to get the same u when I divide the 3rd row by 4 and create with these rows $0$ for the 1st, 2nd and the 4th row?

Is this solution correct, can you see any mistakes I made?

I really appreciate your answer!!!

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You developed along the first column, not the first row. The calculation might be simpler if you developed along the first row. If you divide the 3rd row by $4$, that changes the value of the determinant --- it divides the determinant by $4$, also. –  Gerry Myerson Jan 20 '13 at 9:55

1 Answer 1

up vote 0 down vote accepted

Your computation is correct. If $\vert A\vert =35$, then $\displaystyle u=\frac{4055}{5052}$.

If you divide the third row by $4$, then the determinant of the matrix you get will be $\displaystyle \frac{1}{4}\vert A\vert$.

Finally, if I understand correctly what you mean with "create $0$'s on the other rows", then that, by itself, won't change the determinant. Howerever if you first divide the third row by $4$, you must not forget to multiply by $4$ at the end of your calculations to get the correct determinant of $A$.

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