Problem 34 - College Algebra

Question

In Exercises \(27-36,\) find (if possible): \(\begin{array}{llll}\text { a. } A B & \text { and } & \text { b. } B A\end{array}\) $$ A=\left[\begin{array}{ll} 2 & 4 \\ 3 & 1 \\ 4 & 2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 3 & 2 & 0 \\ -1 & -3 & 5 \end{array}\right] $$

Step-by-Step Solution

Answer

The matrix product AB is \[ \left[ \begin{array}{ccc} 2 & -8 & 20 \ 7 & 1 & 5 \ 8 & 2 & 10 \end{array} \right] \] but BA cannot be found as the number of rows in A is not equal to the number of columns in B.

Step 1: Multiply AB

AB is computed by taking each row of A and dot multiplying it by each column of B. \n For example, the first element of AB is obtained by multiplying the first row of A [2,4] with the first column of B [3,-1], giving 2*3 + 4*(-1) = 2.\n Following the same procedure for all other cells gives the resultant matrix AB = \[ \left[ \begin{array}{ccc} 2 & -8 & 20 \ 7 & 1 & 5 \ 8 & 2 & 10 \end{array} \right] \] \n

Step 2: Multiply BA

Now we compute BA. Notice that now we will take each row of B and dot multiply by each column of A. Specifically, the first element of BA is obtained by multiplying the first row of B [3,2,0] with the first column of A [2, 3, 4], giving \(3*2 + 2*3 + 0*4 = 12\). Following the same procedure for all cells we can obtain the resultant matrix BA. Since it results in a 2x2 matrix and the number of columns in B is not equal to the number of rows in A, in this case, we cannot find the product BA. It is impossible. \n

Key Concepts

Dot Product

The dot product is a fundamental concept in linear algebra, especially when dealing with matrix multiplication. It involves multiplying pairs of numbers from two lists (or vectors) and then summing the results. In the context of matrices, when multiplying two matrices, the dot product is used to compute the elements of the resulting matrix. To perform the dot product between two vectors, you need to multiply the corresponding elements from each vector and add them together:

• If you have vectors \(\mathbf{u} = [u_1, u_2, \ldots, u_n]\) and \(\mathbf{v} = [v_1, v_2, \ldots, v_n]\), the dot product is: \(\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + \ldots + u_nv_n\).

This same principle is applied when multiplying matrices. For example, when computing the product \(AB\), each element of the resulting matrix is the dot product of a row from matrix \(A\) and a column from matrix \(B\). Remember, the dot product provides a single number that represents the interaction of the vectors' dimensions, making it crucial for determining each element in the new matrix.

Matrix Dimensions

Matrix dimensions are crucial to understand because they dictate how matrices can interact with each other through operations like multiplication. Each matrix has dimensions defined as \(\text{rows} \times \text{columns}\), which tells you how many rows and columns it contains. For instance, if matrix \(A\) has dimensions \(3 \times 2\), it means 3 rows and 2 columns. Matrix \(B\), with \(2 \times 3\) dimensions, has 2 rows and 3 columns. These specific dimensions are a green light for multiplying \(A\) by \(B\), because the number of columns in \(A\) (2) matches the number of rows in \(B\) (2).

• The resulting product \(AB\) will have dimensions of \(3 \times 3\).
• The reverse order, \(BA\), is not possible here, as the columns of \(B\) (3) do not match the rows of \(A\) (3).

Understanding matrix dimensions is essential because it sets the rules for what matrix multiplications are feasible, ensuring the operations are mathematically valid.

Non-Commutativity of Matrices

Matrix multiplication is non-commutative, meaning that the order in which you multiply matrices matters a lot. Unlike regular arithmetic multiplication where \(x \cdot y = y \cdot x\), in matrices, \(AB eq BA\) generally. This non-commutative property is tied closely to the matrix dimensions and their arrangement. Consider matrices \(A\) (3x2) and \(B\) (2x3):

• You can compute \(AB\), resulting in a 3x3 matrix, since you have a valid dot product for each element by matching columns of \(A\) with rows of \(B\).
• For \(BA\), it simply isn’t possible because you cannot align the 3 columns of \(B\) appropriately with the 2 rows of \(A\).

This fundamental property reflects the complexity and delicacy involved in matrix operations, illustrating the need for correct sequence and understanding of dimensions before performing such operations. Always remember to verify the order, because even if the resultant matrix exists, it may not hold the same meaning or dimensions.