Question

Given A=[4−231],B=[510−237]A = \left[ \begin{matrix} 4 & -2 \ 3 & 1 \end{matrix} ight], \quad B = \left[ \begin{matrix} 5 & 1 \ 0 & -2 \ 3 & 7 \end{matrix} ight]
 and ﻿C=[−541036]C = \left[ \begin{matrix} -5 & 4 & 1 \ 0 & 3 & 6 \end{matrix} ight]
C=[−50​43​16​]﻿, find each product, if possible. See Examples 5–7. BA

Accepted Answer

Identify the dimensions of matrices B and A. Since B is a 2x3 matrix, it has 2 rows and 3 columns. The matrix A's dimensions must be known or assumed to proceed. For the product BA to be defined, the number of columns in B must equal the number of rows in A. Check the compatibility for multiplication BA. Specifically, if B is 2x3, then A must be a 3xN matrix (where N is any number) for the product BA to be possible. If A is not 3xN, then the product BA is not defined. If the product BA is defined, determine the dimensions of the resulting matrix. The resulting matrix will have the number of rows of B and the number of columns of A, so it will be a 2xN matrix. To compute the product BA, multiply each row of B by each column of A. This involves taking the dot product of the row vector from B with the column vector from A for each element in the resulting matrix. Write the general formula for the element in the ith row and jth column of the product BA as: $$(BA)_{ij} = \sum_{k=1}^{3} B_{ik} 	imes A_{kj}$$ where $$i$$ ranges from 1 to 2 and $$j$$ ranges from 1 to N.

Given $A = [\begin{matrix} 4 & - 2 \\ 3 & 1 \end{matrix}],$ and $C = \(\left\)[ \(\begin{matrix}\) -5 & 4 & 1 \\ 0 & 3 & 6 \(\end{matrix}\) \(\right\)]$ , find each product, if possible. See Examples 5–7. BA

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Key Concepts

Matrix Dimensions and Compatibility

Matrix Multiplication Process

Non-Commutativity of Matrix Multiplication