For the following exercises, use the matrix below to perform...

For the following exercises, use the matrix below to perform the indicated operation on the given matrix. $$ B=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {0} & {1} \\ {0} & {1} & {0}\end{array}\right] $$ Using the above questions, find a formula for $B^{n} .$ Test the formula for $B^{201}$ and $B^{202},$ using a calculator.

For the following exercises, use the matrix below to perform the indicated operation on the given matrix.
$$
B=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {0} & {1} \\ {0} & {1} & {0}\end{array}\right]
$$
Using the above questions, find a formula for $B^{n} .$ Test the formula for $B^{201}$ and $B^{202},$ using a calculator.

Key Concepts

Matrix Exponentiation

Matrix exponentiation involves raising a square matrix to a positive integer power by multiplying the matrix by itself repeatedly. This process is used to study iterative behaviors in linear transformations, and when the matrix has special properties, such as periodicity or symmetry, the calculation can often be simplified using those inherent structures.

Involutory Matrix

An involutory matrix is one that is its own inverse, meaning that when it is multiplied by itself, the result is the identity matrix. This property implies that such matrices have a simple cyclic behavior—for any positive integer exponent, the power of the matrix can be determined based on whether the exponent is even or odd.

Permutation Matrix

A permutation matrix is formed by permuting the rows or columns of an identity matrix, and it represents a rearrangement of coordinates in a linear space. These matrices are used to model the effect of reordering elements and often exhibit properties such as being orthogonal and involutory, which can simplify their exponentiation by leveraging their inherent structure and symmetry.

Transcript

00:04 All right, in this question, we're asked to take matrix b and multiply it by itself enough times to where we can generate a formula for b to the n.

00:14 So we're just going to start off multiplying b by itself a few times until we notice a pattern.

00:19 So we'll first multiply b by itself to get b squared.

00:23 So first row, first column here, we're going to get 1 times 1, 0 times 0, 0.

00:30 Times zero.

00:32 And then second column, we're going to get one time zero, zero times zero, and zero times one.

00:40 And then third column, we're going to get one times zero, zero times one, and zero times zero.

00:45 So that's zero also.

00:47 All right.

00:48 So now we get to do second row first column, zero times one, zero times zero, and one and 1 times 1, so that's 1.

01:06 And then our third column, we get 0 times 0, 0 times 1, and 1 times 0 is 0.

01:13 And then our third row, first column, 0 times 1, 1 times 0, 0 times 0 is 0.

01:21 And then our second column, 0 times 0, 1 times 0, and 0 times 1 times 1 is 0.

01:28 And then our last column we get 0 times 0, 1 times 1, and 0 times 0, which is 1.

01:36 All right, so this right here is b squared.

01:39 So to find b cubed, we're going to then take b squared, 1 -0 -0 -0 -0 -0 -0 -0 -0 -1, and 0 -0 -1, and 0 -1, and 0 -1.

01:48 All right, let's see what we get for.

01:59 B cubes.

02:01 So first row, first column.

02:03 We get 1 times 1, 0 times 0, 0 times 0.

02:09 And second column, 1 times 0, 0 times 0, and 0 times 1.

02:15 And third column, 1 times 0, 0 times 1 and 0 times 0 again.

02:22 And then second row first column, 0 times 1, 1 times 0, 0 times 0, 0.

02:28 And then 0 times 0, 1 times 0, 0 times 1 is 0.

02:35 And then our third column, 0 times 0, 1 times 1, and 0 times 0, 1, and then our third row first column is 0 times 1, 0 times 0, 1 times 0 is 0...

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