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Let
x be the collection of
2×2 matricet in the form
[[1,a],[b,1]] where "
a " and "
b " are real numbers with. the binary operation of addtion defined by
[[1,a],[b,1]]o+[[1,c],[d,1]]=

[[1,a+c],[b+d,1]]. Does it have closune property?

Let $x$ be the collection of $2 \times 2$ matricet in the form $\left[\begin{array}{ll}1 & a \\ b & 1\end{array}\right]$ where

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Write equations of cosine functions using properties

Full solution

Q. Let

x

be the collection of

2 \times 2

matricet in the form

\left[\begin{array}{ll}1 & a \\ b & 1\end{array}\right]

where

Check Closure Under Addition: To check for closure under addition, we need to verify that when we add any two matrices of the given form, the result is also a matrix of the same form.
Choose Arbitrary Matrices: Let's take two arbitrary matrices from our set: $\newline$ Matrix $1$ : $\begin{bmatrix} 1 & a \ b & 1 \end{bmatrix}$ $\newline$ Matrix $2$ : $\begin{bmatrix} 1 & c \ d & 1 \end{bmatrix}$ $\newline$ where $a$ , $b$ , $c$ , and $d$ are real numbers.
Perform Matrix Addition: We add the two matrices using the given binary operation: \begin{bmatrix}1 & a\b & 1\end{bmatrix} \oplus \begin{bmatrix}1 & c\d & 1\end{bmatrix} = \begin{bmatrix}1 & a+c\b+d & 1\end{bmatrix}
Verify Resulting Matrix: The resulting matrix after addition is $\left[\begin{array}{cc}1 & a+c \ b+d & 1\end{array}\right]$ . Since $a+c$ and $b+d$ are sums of real numbers, they are also real numbers.
Closure Under Addition Confirmed: The resulting matrix is of the same form as the original matrices, with the top-right and bottom-left elements being real numbers. This means that the set is closed under the given operation of addition.

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