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

[[1,a+c],[b+d,1]]

Let $x$ be the collection of $2 \times 2$ matrices 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

matrices in the form

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

where

Understand binary operation: Understand the binary operation defined for the collection of $2\times2$ matrices. $\newline$ The binary operation $o+$ is defined for the collection of matrices in the form $[[1,a],[b,1]]$ , where $a$ and $b$ are real numbers. The operation is defined as follows: $\newline$ $[[1,a],[b,1]] \, o+ \, [[1,c],[d,1]] = [[1,a+c],[b+d,1]]$ $\newline$ This operation is essentially matrix addition, where the elements in the corresponding positions are added together.
Perform operation on matrices: Perform the binary operation $o+$ on the given matrices. $\newline$ To find the result of the operation, we simply add the corresponding elements of the two matrices: $\newline$ $\begin{bmatrix}1 & a\ b & 1\end{bmatrix} o+ \begin{bmatrix}1 & c\ d & 1\end{bmatrix} = \begin{bmatrix}1+1 & a+c\ b+d & 1+1\end{bmatrix}$
Simplify resulting matrix: Simplify the resulting matrix. $\newline$ Since $1+1$ is always $2$ , we can simplify the resulting matrix to: $\newline$ \left[\begin{array}{cc}\(\newline\)\(2\) & a+c (\newline\) b+d & \(2\)\(\newline\)\end{array}\right]
Check matrix form: Check if the resulting matrix is still in the form of the collection. The resulting matrix $\left[\begin{array}{cc} 2 & a+c \ b+d & 2 \end{array}\right]$ is not in the form $\left[\begin{array}{cc} 1 & a \ b & 1 \end{array}\right]$ because the diagonal elements are $2$ instead of $1$ . This means that the resulting matrix is not in the same collection as defined by the problem.

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