We’ve seen that natural numbers are closed under addition. Determine the closure of natural numbers under the other three operat
ions, and give examples to support your answers.
1 answer:
Natural numbers are not closed under subtraction. In fact, any operation has no solution using natural numbers.
Natural numbers are closed under multiplication. In fact, any operation means to sum
by itself
times, and we already know that natural numbers are closed under sum.
Natural numbers are not closed under division. In fact, if doesn't divide
, any operation
has no solution using natural numbers.
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The question is incomplete. The complete question is as follows:
Solve for X. Assume X is a 2x2 matrix and I denotes the 2x2 identity matrix. Do not use decimal numbers in your answer. If there are fractions, leave them unevaluated.
· X·
=<em>I</em>.
First, we have to identify the matrix <em>I. </em>As it was said, the matrix is the identiy matrix, which means
<em>I</em> =
So, · X·
=
Isolating the X, we have
X·=
-
Resolving:
X·=
X·=
Now, we have a problem similar to A.X=B. To solve it and because we don't divide matrices, we do X=A⁻¹·B. In this case,
X=⁻¹·
Now, a matrix with index -1 is called Inverse Matrix and is calculated as: A . A⁻¹ = I.
So,
·
=
9a - 3b = 1
7a - 6b = 0
9c - 3d = 0
7c - 6d = 1
Resolving these equations, we have a=; b=
; c=
and d=
. Substituting:
X= ·
Multiplying the matrices, we have
X=