QuestionFind the dimension of the product of matrices sized , , and .
Studdy Solution
STEP 1
Assumptions1. We are given three matrices with dimensions , , and respectively.
. We are asked to find the dimension of the final product of these three matrices.
3. The multiplication of matrices is associative, which means the way in which we group the matrices does not change the final product.
4. The product of two matrices is defined only if the number of columns in the first matrix is equal to the number of rows in the second matrix.
STEP 2
First, we need to understand the rule for matrix multiplication. The product of two matrices A and B (denoted as AB) is defined if and only if the number of columns in A is equal to the number of rows in B. The resulting matrix will have the same number of rows as A and the same number of columns as B.
STEP 3
We can multiply the first two matrices because the number of columns in the first matrix () is equal to the number of rows in the second matrix (). The resulting matrix will have the same number of rows as the first matrix (3) and the same number of columns as the second matrix (5). So, the dimension of the product of the first two matrices is .
STEP 4
Now, we can multiply this resulting matrix with the third matrix because the number of columns in the resulting matrix () is equal to the number of rows in the third matrix (). The final product will have the same number of rows as the resulting matrix (3) and the same number of columns as the third matrix (2).
STEP 5
So, the dimension of the final product of the three matrices is .
The dimension of the final product of the three matrices is .
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