Chapter 2: Q15E (page 184)

Let
$A = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$
Find a formula for $A^{n}$ , whenever n is a positive integer.

Short Answer

Expert verified

$A^{n} = (\begin{matrix} 1 & n \\ 0 & 1 \end{matrix})$

Step by step solution

Step 1

Given:

$A = (\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix})$

$A^{n}$ is the product of A with itself (n - 1) times (n factors in total)

A is a matrix Their product of A with itself is then also a 2 x 2 matrix.

The (i,j) th element of the product AB is the sum of the products of each element in the ith row of A with the corresponding element in the j th column of B .

Step 2

Letus first determine $A^{2}$

$A^{2} = A A = [\begin{array}{ll} 1 1 \\ 0 1 \end{array}] [\begin{array}{ll} 1 1 \\ 0 1 \end{array}] = [\begin{array}{ll} 1 (1) + 1 (0) 1 (1) + 1 (1) \\ 0 (1) + 1 (0) 0 (1) + 1 (1) \end{array}] = [\begin{array}{ll} 1 2 \\ 0 1 \end{array}]$

Next determine $A^{3}$

$A^{3} = A^{2} A = [\begin{array}{ll} 1 1 \\ 0 1 \end{array}] [\begin{array}{ll} 1 1 \\ 0 1 \end{array}] = [\begin{array}{ll} 1 (1) + 2 (0) 1 (1) + 2 (1) \\ 0 (1) + 1 (0) 0 (1) + 1 (1) \end{array}] = [\begin{array}{ll} 1 3 \\ 0 1 \end{array}]$

Next determine $A^{4}$

$A^{4} = A^{3} A = [\begin{array}{ll} 1 1 \\ 0 1 \end{array}] [\begin{array}{ll} 1 1 \\ 0 1 \end{array}] = [\begin{array}{ll} 1 (1) + 3 (0) 1 (1) + 3 (1) \\ 0 (1) + 1 (0) 0 (1) + 1 (1) \end{array}] = [\begin{array}{ll} 1 4 \\ 0 1 \end{array}]$

We then note that the matrix has the elements $A^{n}$ on the diagonal, 0 in the bottom left corner and n in the upper right corner.

data-custom-editor="chemistry" $A^{n} = [\begin{matrix} 1 & n \\ 0 & 1 \end{matrix}]$

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Let
$A = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$
Find a formula for $A^{n}$ , whenever n is a positive integer.

Short Answer

Step by step solution

Step 1

Step 2

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Let A=1101 Find a formula for An, whenever n is a positive integer.

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Step by step solution

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Step 2

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Most popular questions from this chapter

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Let
$A = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$
Find a formula for $A^{n}$ , whenever n is a positive integer.