79471 If the product \(a b c=1\), then the value of the
determinant \(\left|\begin{array}{ccc}-\mathbf{a}^{2} & \mathbf{a b} & \mathbf{a c} \\ \mathbf{b a} & -\mathbf{b}^{2} & \mathbf{b c} \\ \mathbf{a c} & \mathbf{b c} & -\mathbf{c}^{2}\end{array}\right|\) is

79472 If \(a, b\) and \(c\) are distinct reals and the
\(\operatorname{determinant}\left|\begin{array}{lll}
\mathbf{a}^{3}+1 & \mathbf{a}^{2} & \mathbf{a} \\ \mathbf{b}^{3}+1 & \mathbf{b}^{2} & \mathbf{b} \\ \mathbf{c}^{3}+1 & \mathbf{c}^{2} & \mathbf{c} \end{array}\right|=\mathbf{0},\)
then the product abc is

79473 If
\(\left|\begin{array}{ccc} \mathbf{x}+\mathbf{y}+\mathbf{2 z} & \mathbf{x} & \mathbf{y} \\ \mathbf{z} & \mathbf{y}+\mathbf{z}+\mathbf{2 x} & \mathbf{y} \\ \mathbf{z} & \mathbf{x} & \mathbf{z}+\mathbf{x}+\mathbf{2 y} \end{array}\right|\)
\(=k(x+y+z)^{3}\), then the value of \(k\) is

79441 If \(A\) and \(B\) are two matrices such that rank of \(A=m\) and \(\operatorname{rank}\) of \(B=n\), then

79471 If the product \(a b c=1\), then the value of the
determinant \(\left|\begin{array}{ccc}-\mathbf{a}^{2} & \mathbf{a b} & \mathbf{a c} \\ \mathbf{b a} & -\mathbf{b}^{2} & \mathbf{b c} \\ \mathbf{a c} & \mathbf{b c} & -\mathbf{c}^{2}\end{array}\right|\) is

79472 If \(a, b\) and \(c\) are distinct reals and the
\(\operatorname{determinant}\left|\begin{array}{lll}
\mathbf{a}^{3}+1 & \mathbf{a}^{2} & \mathbf{a} \\ \mathbf{b}^{3}+1 & \mathbf{b}^{2} & \mathbf{b} \\ \mathbf{c}^{3}+1 & \mathbf{c}^{2} & \mathbf{c} \end{array}\right|=\mathbf{0},\)
then the product abc is

79473 If
\(\left|\begin{array}{ccc} \mathbf{x}+\mathbf{y}+\mathbf{2 z} & \mathbf{x} & \mathbf{y} \\ \mathbf{z} & \mathbf{y}+\mathbf{z}+\mathbf{2 x} & \mathbf{y} \\ \mathbf{z} & \mathbf{x} & \mathbf{z}+\mathbf{x}+\mathbf{2 y} \end{array}\right|\)
\(=k(x+y+z)^{3}\), then the value of \(k\) is

79441 If \(A\) and \(B\) are two matrices such that rank of \(A=m\) and \(\operatorname{rank}\) of \(B=n\), then

79471 If the product \(a b c=1\), then the value of the
determinant \(\left|\begin{array}{ccc}-\mathbf{a}^{2} & \mathbf{a b} & \mathbf{a c} \\ \mathbf{b a} & -\mathbf{b}^{2} & \mathbf{b c} \\ \mathbf{a c} & \mathbf{b c} & -\mathbf{c}^{2}\end{array}\right|\) is

79472 If \(a, b\) and \(c\) are distinct reals and the
\(\operatorname{determinant}\left|\begin{array}{lll}
\mathbf{a}^{3}+1 & \mathbf{a}^{2} & \mathbf{a} \\ \mathbf{b}^{3}+1 & \mathbf{b}^{2} & \mathbf{b} \\ \mathbf{c}^{3}+1 & \mathbf{c}^{2} & \mathbf{c} \end{array}\right|=\mathbf{0},\)
then the product abc is

79473 If
\(\left|\begin{array}{ccc} \mathbf{x}+\mathbf{y}+\mathbf{2 z} & \mathbf{x} & \mathbf{y} \\ \mathbf{z} & \mathbf{y}+\mathbf{z}+\mathbf{2 x} & \mathbf{y} \\ \mathbf{z} & \mathbf{x} & \mathbf{z}+\mathbf{x}+\mathbf{2 y} \end{array}\right|\)
\(=k(x+y+z)^{3}\), then the value of \(k\) is

79441 If \(A\) and \(B\) are two matrices such that rank of \(A=m\) and \(\operatorname{rank}\) of \(B=n\), then

79471 If the product \(a b c=1\), then the value of the
determinant \(\left|\begin{array}{ccc}-\mathbf{a}^{2} & \mathbf{a b} & \mathbf{a c} \\ \mathbf{b a} & -\mathbf{b}^{2} & \mathbf{b c} \\ \mathbf{a c} & \mathbf{b c} & -\mathbf{c}^{2}\end{array}\right|\) is

79472 If \(a, b\) and \(c\) are distinct reals and the
\(\operatorname{determinant}\left|\begin{array}{lll}
\mathbf{a}^{3}+1 & \mathbf{a}^{2} & \mathbf{a} \\ \mathbf{b}^{3}+1 & \mathbf{b}^{2} & \mathbf{b} \\ \mathbf{c}^{3}+1 & \mathbf{c}^{2} & \mathbf{c} \end{array}\right|=\mathbf{0},\)
then the product abc is

79473 If
\(\left|\begin{array}{ccc} \mathbf{x}+\mathbf{y}+\mathbf{2 z} & \mathbf{x} & \mathbf{y} \\ \mathbf{z} & \mathbf{y}+\mathbf{z}+\mathbf{2 x} & \mathbf{y} \\ \mathbf{z} & \mathbf{x} & \mathbf{z}+\mathbf{x}+\mathbf{2 y} \end{array}\right|\)
\(=k(x+y+z)^{3}\), then the value of \(k\) is

79441 If \(A\) and \(B\) are two matrices such that rank of \(A=m\) and \(\operatorname{rank}\) of \(B=n\), then