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Q29E

Expert-verifiedFound in: Page 165

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

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

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

** **

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

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