Question

The determinant $\begin{vmatrix} 1+a+x & a+y & a+z \\ b+x & 1+b+y & b+z \\ c+x & c+y & 1+c+z \end{vmatrix} =$

• A. $(1+a+b+c)(1+x+y+z)-3(ax+by+cz)$
• B. $a(x+y)+b(y+z)+c(z+x)-(xy+yz+zx)$
• C. $x(a+b)+y(b+c)+z(c+a)-(ab+bc+ca)$
• D. none of these

Answer 1

Answer: (A)

$(1+a+b+c)(1+x+y+z)-3(ax+by+cz)$

Answer 2

Express the given matrix as $I+u v^T+u'v'^T$. Use the determinant formula for rank–two updates:

$$\det(M)=\det\Big(I_2+\begin{pmatrix} v^T u & v^T u'\\ v'^T u & v'^T u'\end{pmatrix}\Big)$$

to obtain:

$$(1+a+b+c)(1+x+y+z)-3(ax+by+cz).$$