To determine which matrices a \(2 \times 3\) matrix can be multiplied by, we look at the rules of matrix multiplication. Specifically, if matrix \(A\) (with dimensions \(m \times n\)) is multiplied by matrix \(B\) (with dimensions \(p \times q\)), then the multiplication is possible if and only if \(n = p\). The result will be a matrix of dimensions \(m \times q\).
Given a \(2 \times 3\) matrix (let's denote it as \(A\)):
- Dimensions of \(A\): \(2\) rows and \(3\) columns (\(m = 2\) and \(n = 3\)).
To find matrices that can be multiplied by \(A\), we need matrices \(B\) such that the number of columns in \(A\) (which is \(3\)) equals the number of rows in \(B\) (which we denote as \(p\)).
Now let's evaluate the options one by one:
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\(2 \times 2\) - This matrix has \(2\) rows and \(2\) columns. Here, \(p = 2\), which does not equal \(n = 3\). Not valid.
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\(3 \times 12\) - This matrix has \(3\) rows and \(12\) columns. Here, \(p = 3\), which equals \(n = 3\). Valid. Multiplication possible.
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\(2 \times 12\) - This matrix has \(2\) rows and \(12\) columns. Here, \(p = 2\), which does not equal \(n = 3\). Not valid.
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\(2 \times 3\) - This matrix has \(2\) rows and \(3\) columns. Here, \(p = 2\), which does not equal \(n = 3\). Not valid.
In summary, the \(2 \times 3\) matrix can only be multiplied by the \(3 \times 12\) matrix from the options given.
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