# If a matrix and its determinant given and another matrix also given how to obtain the second matrix determinant?

If the matrix $$A = \pmatrix{row1 \\ row2\\row3}\ and \left|\begin{array}[ccc]\\ A \end{array}\right| =10$$

and matrix

$$B = \pmatrix{2row1+row2-row3 \\ 2row3\\5row2}\ \$$ then find $$\\\left|\begin{array}[ccc]\\ B \end{array}\right| = ?$$

I am stuck at this. I know that subtracting a multiple of one row from another row does not change determinate (A). Also, if we do permutation of rows 1 time the sign will be negative, but I do not know if this information is useful here or not.

I need help with this one and after reading StackTD hints B, will be: $$B = \pmatrix{2row1\\2row3\\5row2}\ \$$ I know the determinant of B will in negative sign because of rows swap, but I couldn't obtain B determinant.

Use properties of determinants:

• the determinant is linear in each row/column;
• a determinant with two identical rows is $$0$$;
• swapping two rows changes the sign of the determinant.

Now start with linearity and follow up (I write $$A_i$$ for the $$i$$th row of the original matrix $$A$$): $$\begin{vmatrix} 2A_1+A_2-A_3 \\ 2A_3 \\ 5A_2 \end{vmatrix} = \begin{vmatrix} 2A_1 \\ 2A_3 \\ 5A_2 \end{vmatrix}+\begin{vmatrix} A_2 \\ 2A_3 \\ 5A_2 \end{vmatrix}+\begin{vmatrix} -A_3 \\ 2A_3 \\ 5A_2 \end{vmatrix} = \ldots$$

Addition after comment: $$\begin{vmatrix} \color{blue}{2}A_1 \\ \color{green}{2}A_3 \\ \color{red}{5}A_2 \end{vmatrix}=\color{blue}{2}\cdot\color{green}{2}\cdot\color{red}{5}\cdot\begin{vmatrix} A_1 \\ \color{purple}{A_3} \\ \color{purple}{A_2} \end{vmatrix}=\ldots$$

• thanks for explaining that to me, now i updated the subject with matrix B but i couldn't obtain B determinant is that right ?, and how we obtain b determinant ? Dec 12, 2018 at 12:50
• Linearity also allows you to 'extract' the multiples (in each row!); then notice the swap of two rows. See updated answer. Dec 12, 2018 at 12:53
• thank you again for explaining to me now i have more understading of determinant properties because of your help, and from what you have told me the determinant of B will be -20 right ? Dec 12, 2018 at 13:00
• Almost, $-20$ times the determinant $A$, so...? Dec 12, 2018 at 13:27
• so then the determinant of b will be -10 ? Dec 12, 2018 at 13:56