Problem 51 - A Graphical Approach to College Algebra

Question

If possible, find \(A B\) and \(B A.\) $$A=\left[\begin{array}{rrr} 5 & -7 & 2 \\ 0 & 1 & 5 \end{array}\right], \quad B=\left[\begin{array}{rrr} 9 & 8 & 7 \\ 1 & -1 & -2 \end{array}\right]$$

Step-by-Step Solution

Answer

Both AB and BA are not defined due to incompatible dimensions.

Step 1: Determine Compatibility of AB

To multiply matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, matrix \(A\) is 2x3 and matrix \(B\) is also 2x3. Since the number of columns in \(A\) does not match the number of rows in \(B\), the matrix product \(AB\) is not defined.

Step 2: Determine Compatibility of BA

Now, let's check the compatibility for \(BA\). Matrix \(B\) is 2x3, and matrix \(A\) is 2x3. Again, the number of columns in \(B\) does not match the number of rows in \(A\), so the matrix product \(BA\) is also not defined.

Step 3: Conclusion

Since neither \(AB\) nor \(BA\) are compatible for multiplication based on their dimensions, neither product can be computed.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is key to grasping matrix multiplication. A matrix's dimensions give us a sense of its size and shape, described as 'rows x columns'. For example, matrix \(A\) has dimensions 2x3, meaning it has 2 rows and 3 columns. Similarly, matrix \(B\) also has dimensions 2x3, which is typical for many matrix problems.These dimensions are critical when determining if matrix products are possible. Knowing dimensions helps avoid errors and evaluate compatibility quickly. Always start by checking dimensions before attempting any matrix operation.

Matrix Compatibility

Matrix compatibility is an essential concept in matrix multiplication. It determines whether two matrices can be multiplied together. To be compatible, the number of columns in the first matrix must equal the number of rows in the second matrix. This condition ensures that each element in a row of the first matrix can be paired with each element in a column of the second matrix.In the given example, neither \(AB\) nor \(BA\) meet this condition:

• Matrix \(A\) has 3 columns, and matrix \(B\) has only 2 rows; thus, \(AB\) is not compatible.
• Matrix \(B\) also has 3 columns, and matrix \(A\) has only 2 rows; therefore, \(BA\) is similarly incompatible.

Understanding compatibility helps prevent computation errors and saves time during problem-solving.

Matrix Product

The matrix product is the result of multiplying two compatible matrices. It involves a systematic process where the corresponding entries are combined to form a new matrix. However, if the matrices are not compatible for multiplication, as in our current problem, the product cannot be computed. This highlights the importance of compatibility for yielding a valid matrix product. Knowing how to check compatibility first will save you from unnecessary calculations and errors. Remember, a successful matrix product yields a new matrix with dimensions determined by the number of rows of the first matrix and columns of the second.