Suppose A, B, and C are matrices, and C = AB is defined. Then C has the same number of columns as A and the same number of rows as B.
To understand why C, the product of matrices A and B, has the same number of columns as A and the same number of rows as B, we need to examine matrix multiplication
To understand why C, the product of matrices A and B, has the same number of columns as A and the same number of rows as B, we need to examine matrix multiplication.
When two matrices are multiplied, their dimensions play a significant role in determining the size of the resulting matrix. Let’s consider the dimensions of the matrices involved.
Matrix A has dimensions m x n, where m and n represent the number of rows and columns, respectively. Hence, A has m rows and n columns.
Matrix B has dimensions n x p, where p represents the number of columns. We can observe that the number of columns of B matches the number of rows of A, which is denoted by n.
Now, when we perform matrix multiplication between A and B, the resulting matrix, C, will have dimensions m x p.
Why does C have m rows and p columns?
The number of rows in C corresponds to the number of rows in A, which is given by m.
The number of columns in C is determined by the number of columns in B, which is p.
Therefore, based on the dimensions of A and B, we find that matrix C will have m rows (the same number as A) and p columns (the same number as B). Thus, C = AB is defined as a matrix with the same number of columns as A and the same number of rows as B.
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