Compute \(det{\rm{ }}{B^4}\), where \(B = \left[ {\begin{aligned}{{}{}}1&0&1\\1&1&2\\1&2&1\end{aligned}} \right]\).

If A and B aresquare matrices, then thedeterminant of the product matrix AB is equal to the product of determinant of A and determinant of B.

\(\det AB = \left( {\det A} \right)\left( {\det B} \right)\)

If the matrices A and B are the same, then the general form is

\(\det {A^n} = {\left( {\det A} \right)^n}\).

Compute the determinant of the matrix as shown below:

\(\begin{aligned}{}\det B &= \left| {\begin{aligned}{{}{}}1&0&1\\1&1&2\\1&2&1\end{aligned}} \right|\\ &= 1 \cdot \left( {1\left( 1 \right) - 2\left( 2 \right)} \right) - 0\left( {1\left( 1 \right) - 2\left( 1 \right)} \right) + 1\left( {1\left( 2 \right) - 1\left( 1 \right)} \right)\\ &= - 3 - 0 + 1\\ &= - 2\end{aligned}\)

Thus, \(\det B = - 2\).

Obtain the value of\(\det {\rm{ }}{B^4}\)using themultiplicative property.

\(\begin{aligned}{}\det {\rm{ }}{B^4} &= {\left( {\det B} \right)^4}\\ &= {\left( { - 2} \right)^4}\\ &= 16\end{aligned}\)

Thus, \(\det {\rm{ }}{B^4} = 16\).