Problem 44 - College Algebra and Calculus: An Applied Approach

Question

Find (a) \(A B\), (b) \(B A\), and, if possible, (c) \(A^{2}\). (Note: \(A^{2}=A A\).) $$ A=\left[\begin{array}{llll} 3 & 2 & 1 & 0 \end{array}\right], B=\left[\begin{array}{l} 2 \\ 3 \\ 1 \\ 0 \end{array}\right] $$

Step-by-Step Solution

Answer

(a) \(AB = 13\), (b) BA does not exist, (c) \(A^{2}\) does not exist.

Step 1: Find AB

The number of columns in matrix A equals the number of rows in matrix B, so A and B are compatible for multiplication. Each entry in the resulting matrix is the sum of the products of corresponding entries from A and B.\n\[AB = \left[ \begin{array}{llll} 3 & 2 & 1 & 0 \end{array} \right] \cdot \left[ \begin{array}{l} 2 \ 3 \ 1 \ 0 \end{array} \right] = (3*2 + 2*3 + 1*1 + 0*0) = 13\]

Step 2: Find BA

The number of columns in B does not equal the number of rows in A which means they are not compatible for multiplication. The product BA does not exist.

Step 3: Find \(A^{2}\)

The operation \(A^{2}\) (or \(AA\)) requires the number of columns in the first matrix A to equal the number of rows in the second matrix A. This is not the case here, so \(A^{2}=AA\) does not exist.

Key Concepts

Matrix Compatibility

In order to multiply two matrices, they must be compatible with each other. This means the number of columns in the first matrix must match the number of rows in the second matrix. Compatibility is a key step before even attempting matrix multiplication. In the given exercise, matrix \( A \) has one row and four columns, while matrix \( B \) has four rows and one column. This setup makes them compatible for multiplication as the four columns of \( A \) align perfectly with the four rows of \( B \). Once compatibility is verified, you can proceed with calculating the matrix product. Remember, it's essential to always check compatibility first to avoid errors when performing matrix operations.

Matrix Product

When two matrices are compatible, we can calculate their matrix product. The result of multiplying two matrices, such as \( AB \), is a new matrix. Each element in the resulting matrix is calculated by taking the sum of the products of corresponding elements from the rows of the first matrix and the columns of the second matrix.

The matrix product \( AB \) in the exercise is derived from multiplying matrix \( A = [3 \, 2 \, 1 \, 0] \) with matrix \( B = \begin{bmatrix} 2 \ 3 \ 1 \ 0 \end{bmatrix} \). This results in:

• First, multiply the corresponding entries: \(3 \times 2\), \(2 \times 3\), \(1 \times 1\), and \(0 \times 0\).
• Sum these products: \(6 + 6 + 1 + 0\).
• Thus, the matrix product is \(13\).

Calculating matrix products involves both correct setup and execution of multiplication and addition, meticulously following the order to ensure accuracy.

Non-Existence of Matrix Product

Sometimes, matrices cannot be multiplied together because they aren't compatible. This concept of non-existence arises when the required conditions for matrix multiplication aren't met. For instance, to find \( BA \), the number of columns in the first matrix \( B \) (which has one column) does not match the number of rows in \( A \) (which has one row).

Similarly, attempting to find \( A^2 \) involves the operation \( AA \). This requires the number of columns in the first \( A \) to equal the number of rows in the second \( A \). However, matrix \( A \) has one row and four columns, which does not meet the criteria necessary for its own multiplication by another \( A \).

In these two scenarios, one for \( BA \) and another for \( A^2 \), the required dimensions do not match. As a result, these matrix products do not exist. Understanding these limitations is vital in matrix operations, ensuring calculations are valid and meaningful.