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Expert-verified Found in: Page 165 ### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384 # Compute $$det{\rm{ }}{B^4}$$, where B = \left[ {\begin{aligned}{{}{}}1&0&1\\1&1&2\\1&2&1\end{aligned}} \right].

The value is $$\det {\rm{ }}{B^4} = 16$$.

See the step by step solution

## Step 1: Write the multiplicative property

If A and B are square matrices, then the determinant 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

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

## Step 2: Find the determinant of the matrix

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 the multiplicative property.

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

Thus, $$\det {\rm{ }}{B^4} = 16$$. ### Want to see more solutions like these? 