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# If $A$ is order $m \times n$, $B$ is of order $3 \times p$ & $C$ is of order $2 \times 4$. If $AB + C$ is defined, find $m,\;n,{\text{ }}\& {\text{ }}p$

Last updated date: 13th Jun 2024
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Hint: In this question, the orders of specifics are given. We know that, if $A$ is a matrix of order $m \times n$ and $B$ is a matrix of order of $n \times p$. The matrix product $AB$ is defined to be an $m \times p$ matrix. The two matrices must be the same size, that is the rows must match in size, and the columns must match in size for matrix addition.

It is given that, $A$ is of order $m \times n$, $B$ is of order $3 \times p$& $C$ is of order $2 \times 4$.
Also given that, $AB + C$ is defined.
We need to find out $m,\;n,{\text{ }}\& {\text{ }}p$.
We know that the number of columns of the first matrix must equal the number of rows of the second matrix. If $A$ is a matrix of order $m \times n$ and $B$ is a matrix of order of $n \times p$.The matrix product $AB$ is defined to be an $m \times p$ matrix.
For given matrix, $A$ is of order $m \times n$, $B$ is of order $3 \times p$, thus $AB$ is of order $m \times p$ and $n$ must be equal to $3$.
For matrix addition two matrices must be the same size, that is the rows must match in size, and the columns must match in size.
Given that, $AB + C$ is defined, so the order of $AB$ and $C$ are equal.
Therefore,$m = 2,p = 4$

Hence we get, $m = 2,n = 3,p = 4$.

Note: In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix.
When we do multiplication:
The number of columns of the first matrix must equal the number of rows of the second matrix.
And the result will have the same number of rows as the first matrix, and the same number of columns as the second matrix.