# True or False: Let $\begin{vmatrix}a & p & x\\b & q & y\\c & r & z\end{vmatrix}$ = 16, then $\Delta_1=\begin{vmatrix}p+x & a+x & a+p\\q+y & b+y & b+q\\r+z & c+z & c+r\end{vmatrix}$ = 32

Toolbox:
• Every square matrix can be associated to an expression or a number which is known as its determinant.
• The determinant of matrix $A=\begin{bmatrix}a_1 & b_1 & c_1\\a_2 & b_2 & c_2\\a_3 & b_3 &c_3\end{bmatrix}$
• $|A|=a_1\begin{vmatrix}b_2 & c_2\\b_3 & c_3\end{vmatrix}-b_1\begin{vmatrix}a_2 & c_2\\a_3 & c_3\end{vmatrix}+c_1\begin{vmatrix}a_2 & b_2\\a_3 & b_3\end{vmatrix}$
Given $\Delta=\begin{vmatrix}a & p &x\\b & q &y\\c &r &z\end{vmatrix}$ then $\Delta_1=\begin{vmatrix}p+x& a+x &a+p\\q+y & b+y & b+q\\r+z & c+z & c+r\end{vmatrix}=16$

Consider $\Delta_1=\begin{vmatrix}p+x& a+x &a+p\\q+y & b+y & b+q\\r+z & c+z & c+r\end{vmatrix}$
A determinant can be expressed as sum of two(or more)determinants.
Therefore $\Delta_1=\begin{vmatrix}p & a& a\\q &b & b\\r & c& c\end{vmatrix}+\begin{vmatrix}x & x& p\\y & y &q\\z & z & r\end{vmatrix}$
But we know if two rows or columns are identical then the value of the determinant is zero.
$\Delta_1=\Delta_a+\Delta_b$
In $\Delta_a=C_2$ and $C_3$ are identical and in $\Delta_b=C_1$ and $C_2$ are identical.
Therefore $\Delta_1=0+0$
$\qquad\qquad\quad=0$
Hence False.